ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o...
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![Page 1: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/1.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia diferentiala liniara cu coeficienti constanti
1 Ecuatia diferentiala liniara cu coeficienti constantiEcuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
2 Ecuatia Euler- Cauchy
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 2: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/2.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia diferentiala liniara cu coeficienticonstanti
February 9, 2011
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 3: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/3.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Forma generala
Consideram ecuatia diferentiala liniara cu coeficienti constanti
y (n) + a1y (n−1) + . . .+ any = f (x), x ∈ (a,b) ⊂ R (1)
unde f este o functie continua si fie ecuatia omogena asociata
y (n) + a1y (n−1) + . . .+ any = 0. (2)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 4: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/4.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Forma generala
Consideram ecuatia diferentiala liniara cu coeficienti constanti
y (n) + a1y (n−1) + . . .+ any = f (x), x ∈ (a,b) ⊂ R (1)
unde f este o functie continua si fie ecuatia omogena asociata
y (n) + a1y (n−1) + . . .+ any = 0. (2)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 5: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/5.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Solutia generala a ecuatei diferentiale liniare
TeoremaSolutia generala a ecuatiei diferentiale liniare (1) este
y(x) = c1y1(x) + . . .+ cnyn(x) + yp (3)
unde y1, . . . , yn sunt solutii liniar independente ale ecuatieiomogene (2), iar yp este o solutie particulara.
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 6: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/6.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Ecuatia caracteristica
Cautam solutii de forma
y(x) = eλx .
Prin derivare si înlocuire în ecuatia omogena obtinem ecuatiacare se numeste caracteristica
λn + a1λn−1 + . . .+ an = 0. (4)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 7: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/7.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Ecuatia caracteristica
Cautam solutii de forma
y(x) = eλx .
Prin derivare si înlocuire în ecuatia omogena obtinem ecuatiacare se numeste caracteristica
λn + a1λn−1 + . . .+ an = 0. (4)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 8: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/8.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
I. Radacini reale distincteDaca λ1, . . . , λn ∈ R distincte sunt radacinile ecuatieicaracteristice atunci forma generala a solutiei este
y(x) = c1eλ1x + . . .+ cneλnx . (5)
II. Radacini reale multiplePresupunem ca λ = α este o solutie a ecuatiei caracteristice cuordinul de multiplicitate r .Atunci solutia generala va cuprinde în cazul II. urmatoareacombinatie liniara
c1eαx + c2xeαx + . . . cr x r−1eαx , c1, c2, . . . , cr ∈ R (6)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 9: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/9.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
I. Radacini reale distincteDaca λ1, . . . , λn ∈ R distincte sunt radacinile ecuatieicaracteristice atunci forma generala a solutiei este
y(x) = c1eλ1x + . . .+ cneλnx . (5)
II. Radacini reale multiplePresupunem ca λ = α este o solutie a ecuatiei caracteristice cuordinul de multiplicitate r .Atunci solutia generala va cuprinde în cazul II. urmatoareacombinatie liniara
c1eαx + c2xeαx + . . . cr x r−1eαx , c1, c2, . . . , cr ∈ R (6)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 10: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/10.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
I. Radacini reale distincteDaca λ1, . . . , λn ∈ R distincte sunt radacinile ecuatieicaracteristice atunci forma generala a solutiei este
y(x) = c1eλ1x + . . .+ cneλnx . (5)
II. Radacini reale multiplePresupunem ca λ = α este o solutie a ecuatiei caracteristice cuordinul de multiplicitate r .Atunci solutia generala va cuprinde în cazul II. urmatoareacombinatie liniara
c1eαx + c2xeαx + . . . cr x r−1eαx , c1, c2, . . . , cr ∈ R (6)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 11: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/11.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
III. Radacini complexe simplePresupunem ca ecuatia caracteristica admite solutiile simpleλ = α± jβ.Atunci solutia generala cuprinde termenii
c1eαx cos(βx) + c2eαx sin(βx) (7)
IV. Radacini reale complexe multiplePresupunem ca ecuatia caracteristica admite solutiileλ = α± jβ cu ordinul de multiplicitate s.Acestora le corespund 2s solutii linar independente
eαx cos(βx) eαx sin(βx)xeαx cos(βx) xeαx sin(βx)
. . . . . .
xs−1eαx cos(βx) xs−1eαx sin(βx)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 12: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/12.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
III. Radacini complexe simplePresupunem ca ecuatia caracteristica admite solutiile simpleλ = α± jβ.Atunci solutia generala cuprinde termenii
c1eαx cos(βx) + c2eαx sin(βx) (7)
IV. Radacini reale complexe multiplePresupunem ca ecuatia caracteristica admite solutiileλ = α± jβ cu ordinul de multiplicitate s.Acestora le corespund 2s solutii linar independente
eαx cos(βx) eαx sin(βx)xeαx cos(βx) xeαx sin(βx)
. . . . . .
xs−1eαx cos(βx) xs−1eαx sin(βx)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 13: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/13.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
III. Radacini complexe simplePresupunem ca ecuatia caracteristica admite solutiile simpleλ = α± jβ.Atunci solutia generala cuprinde termenii
c1eαx cos(βx) + c2eαx sin(βx) (7)
IV. Radacini reale complexe multiplePresupunem ca ecuatia caracteristica admite solutiileλ = α± jβ cu ordinul de multiplicitate s.Acestora le corespund 2s solutii linar independente
eαx cos(βx) eαx sin(βx)xeαx cos(βx) xeαx sin(βx)
. . . . . .
xs−1eαx cos(βx) xs−1eαx sin(βx)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 14: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/14.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
III. Radacini complexe simplePresupunem ca ecuatia caracteristica admite solutiile simpleλ = α± jβ.Atunci solutia generala cuprinde termenii
c1eαx cos(βx) + c2eαx sin(βx) (7)
IV. Radacini reale complexe multiplePresupunem ca ecuatia caracteristica admite solutiileλ = α± jβ cu ordinul de multiplicitate s.Acestora le corespund 2s solutii linar independente
eαx cos(βx) eαx sin(βx)xeαx cos(βx) xeαx sin(βx)
. . . . . .
xs−1eαx cos(βx) xs−1eαx sin(βx)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 15: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/15.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Exercitii
Rezolvati urmaroarele ecuatii omogene1. y ′′ − 5y ′ + 6y = 02. y ′′ − 9y = 03. y ′′ − y ′ = 04. y ′′ − 2y ′ + 2y = 05. y ′′ + 4y ′ + 13y = 06. y ′′′ − 13y ′′ + 12y ′ = 07. y ′′′ − y ′ = 08. y ′′′ + y = 09. y (4) − 2y ′′ = 010. y ′′′ − 3y ′′ + 3y ′ − y = 0
11. y (n) +n1
y (n−1) +n(n − 1)
2y (n−2) + · · ·+ ny ′ + y = 0
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 16: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/16.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Rezolvati urmatoarele probleme Cauchy
1.
y ′′ − 5y ′ + 4y = 0y(0) = 5y ′(0) = 8
2.
y ′′ + 4y = 0y(0) = 0y ′(0) = 2
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 17: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/17.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 18: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/18.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 19: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/19.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 20: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/20.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 21: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/21.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 22: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/22.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 23: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/23.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem caf contine P(x)eax ,
a nu este radacina a ecuatiei caracteristice,iar P este un polinom de gradul m;
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax . (8)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 24: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/24.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 25: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/25.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 26: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/26.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 27: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/27.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 28: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/28.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 29: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/29.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 30: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/30.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax ,
a este radacina a ecuatiei caracteristice cu ordinul demultiplicitate k ,iar P este un polinom de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax . (9)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 31: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/31.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 32: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/32.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 33: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/33.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 34: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/34.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 35: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/35.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 36: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/36.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 37: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/37.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx),numarul complex a± jb nu este radacina a ecuatieicaracteristice,iar P si Q sunt polinoame de grad cel mult m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)eax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)eax sin(bx). (10)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 38: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/38.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx)
numarul complex a± jb este radacina a ecuatieicaracteristice cu ordinul de multiplicitate k ,iar P si Q sunt polinoame de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)xkeax sin(bx). (11)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 39: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/39.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx)
numarul complex a± jb este radacina a ecuatieicaracteristice cu ordinul de multiplicitate k ,iar P si Q sunt polinoame de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)xkeax sin(bx). (11)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 40: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/40.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx)
numarul complex a± jb este radacina a ecuatieicaracteristice cu ordinul de multiplicitate k ,iar P si Q sunt polinoame de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)xkeax sin(bx). (11)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 41: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/41.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx)
numarul complex a± jb este radacina a ecuatieicaracteristice cu ordinul de multiplicitate k ,iar P si Q sunt polinoame de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)xkeax sin(bx). (11)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 42: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/42.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Presupunem ca
f contine P(x)eax cos(bx), sau Q(x)eax sin(bx)
numarul complex a± jb este radacina a ecuatieicaracteristice cu ordinul de multiplicitate k ,iar P si Q sunt polinoame de gradul m.
Atunci solutia particulara contine
(A0 + A1x + . . .+ Amxm)xkeax cos(bx)+
+(B0 + B1x + . . .+ Bmxm)xkeax sin(bx). (11)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 43: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/43.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Rezolvati urmatoarele ecuatii diferentiale1. y ′′ + y ′ − 2y = 8 sin 2x
2. y ′′ + y ′ − 6y = xe2x
3. y ′′ − y ′ − y = x3 − 6
4.{
y ′′ + 4y = sin xy(0) = y ′(0) = 1
5.d2xdt2 + ω2x = A sin pt ; discutie ω 6= p, ω = p
6. y ′′ − y = 2x sin x
7. y ′′′ + y ′′ = x2 + 1 + 3xex
8. y (4) + y ′′′ = cos x
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 44: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/44.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Metoda variatiei constantelor
Fie ecuatia
y ′′ + a(x)y ′ + b(x)y = f (x), x ∈ I ⊂ R (12)
unde f ,a,b sunt functii continue pe intervalul I.Din teorie, solutia generala este de forma
y(x) = c1y1(x) + c2y2(x) + yp(x)
unde y1, y2 sunt solutii liniar independente ale ecuatieiomogene, iar yp este o solutie particulara.Vom arata ca se pot determina doua functii c1(x), c2(x) astfelca solutia particulara sa fie de forma
yp(x) = c1(x)y1(x) + c2(x)y2(x) (13)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 45: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/45.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Metoda variatiei constantelor
Fie ecuatia
y ′′ + a(x)y ′ + b(x)y = f (x), x ∈ I ⊂ R (12)
unde f ,a,b sunt functii continue pe intervalul I.Din teorie, solutia generala este de forma
y(x) = c1y1(x) + c2y2(x) + yp(x)
unde y1, y2 sunt solutii liniar independente ale ecuatieiomogene, iar yp este o solutie particulara.Vom arata ca se pot determina doua functii c1(x), c2(x) astfelca solutia particulara sa fie de forma
yp(x) = c1(x)y1(x) + c2(x)y2(x) (13)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 46: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/46.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Metoda variatiei constantelor
Fie ecuatia
y ′′ + a(x)y ′ + b(x)y = f (x), x ∈ I ⊂ R (12)
unde f ,a,b sunt functii continue pe intervalul I.Din teorie, solutia generala este de forma
y(x) = c1y1(x) + c2y2(x) + yp(x)
unde y1, y2 sunt solutii liniar independente ale ecuatieiomogene, iar yp este o solutie particulara.Vom arata ca se pot determina doua functii c1(x), c2(x) astfelca solutia particulara sa fie de forma
yp(x) = c1(x)y1(x) + c2(x)y2(x) (13)
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 47: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/47.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Se obtin urmatoarele etapePas 1. Se considera ecuatia
y ′′ + a(x)y ′ + b(x)y = f (x)
Pas 2. Se determina y1, y2 solutii independente ale ecuatieiomogene si formam sistemul{
c′1(x)y1(x) + c′2(x)y2(x) = 0c′1(x)y ′1(x) + c′2(x)y ′2(x) = f (x)
Pas 3. Determinam c′1, c′2 si prin integrare aflam c1, c2.
Pas 4. Solutia particulara este
y(x) = c1(x)y1(x) + c2(x)y2(x).
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 48: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/48.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Ecuatia liniara omogenaMetoda coeficientilor nedeterminatiMetoda variatiei constantelor
Rezolvati prin metoda variatiei constantelor
1. y ′′ + y = tan x
2. y ′′ − 2y ′ + y =ex
x
3. y ′′ = y +1
cos x
4. y ′′ + y =1
sin x
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 49: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/49.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Consideram ecuatia
xny (n) + a1xn−1y (n−1) + . . .+ any = 0. (14)
Facem schimbarea de variabila independenta, pentru x > 0
x = et (15)
Au loc imediat formulele de derivarey ′ =
dydt
dtdx
=dydt
1et =
dydt
1x
y ′′ =d2ydt2
1x2 −
dydt
1x2 .
Procedeul continua. Prin înlocuire în ecuatie se obtine oecuatie cu coeficienti constanti.
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 50: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/50.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Consideram ecuatia
xny (n) + a1xn−1y (n−1) + . . .+ any = 0. (14)
Facem schimbarea de variabila independenta, pentru x > 0
x = et (15)
Au loc imediat formulele de derivarey ′ =
dydt
dtdx
=dydt
1et =
dydt
1x
y ′′ =d2ydt2
1x2 −
dydt
1x2 .
Procedeul continua. Prin înlocuire în ecuatie se obtine oecuatie cu coeficienti constanti.
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 51: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/51.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Consideram ecuatia
xny (n) + a1xn−1y (n−1) + . . .+ any = 0. (14)
Facem schimbarea de variabila independenta, pentru x > 0
x = et (15)
Au loc imediat formulele de derivarey ′ =
dydt
dtdx
=dydt
1et =
dydt
1x
y ′′ =d2ydt2
1x2 −
dydt
1x2 .
Procedeul continua. Prin înlocuire în ecuatie se obtine oecuatie cu coeficienti constanti.
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 52: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/52.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Consideram ecuatia
xny (n) + a1xn−1y (n−1) + . . .+ any = 0. (14)
Facem schimbarea de variabila independenta, pentru x > 0
x = et (15)
Au loc imediat formulele de derivarey ′ =
dydt
dtdx
=dydt
1et =
dydt
1x
y ′′ =d2ydt2
1x2 −
dydt
1x2 .
Procedeul continua. Prin înlocuire în ecuatie se obtine oecuatie cu coeficienti constanti.
Ecuatia diferentiala liniara cu coeficienti constanti
![Page 53: Ecua¸tia diferen¸tiala liniar˘ a cu coeficien¸ti constan¸ti˘ · omogene (2), iar yp este o solu¸tie particulara.˘ Ecua¸tia diferen¸tial˘a liniara cu coeficien¸ti constan¸ti](https://reader033.vdocuments.site/reader033/viewer/2022050718/5e18ba69c0787631ba6dec6b/html5/thumbnails/53.jpg)
Ecuatia diferentiala liniara cu coeficienti constantiEcuatia Euler- Cauchy
Rezolvati urmatoarele ecuatii Euler
1. x2y ′′ + 3xy ′ + y = 0
2. x2y ′′ − xy ′ − 3y = 0
3. x3y ′′′ − 3x2y ′′ + 6xy ′ − 6y = 0.
Ecuatia diferentiala liniara cu coeficienti constanti