differential equations separable examples prepared by vince zaccone for campus learning assistance...
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Differential Equations
Separable Examples
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
0byyt
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Next do some algebra until all the y terms are on one side and t terms are on the other:
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnbyln
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnbyln
Ctlnbyln ee
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnbyln
Ctlnbyln ee
Ctlnb eey
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnbyln
Ctlnbyln ee
Ctlnb eey
Ceybtln
This C is really eC.
Since it is still arbitrary we can still call it C.
Use separation of variables to find the general solution to the given differential equation.
0byyt
First, write the derivative in fraction form. 0bydtdy
t
Next do some algebra until all the y terms are on one side and t terms are on the other:
dttb
ydy
dtbydyt
bydtdy
t
0bydtdy
t
typically you want to leave constants on the t side
Now we can integrate both sides and solve for y.
dttb
ydy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnbyln
Ctlnbyln ee
Ctlnb eey
Ceybtln
btCy
This C is really eC.
Since it is still arbitrary we can still call it C.
22 yy et2dtdy
yet2yy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
dtt2dyey
dtt2e
dyy
2
2
y
y
We’ll need to use a substitution for the y integral:
dudyyy2dydu
yu212
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
dtt2dyey
dtt2e
dyy
2
2
y
y
We’ll need to use a substitution for the y integral:
dudyyy2dydu
yu212
22 y21u
21u
21y eeduedyey
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
dtt2dyey
dtt2e
dyy
2
2
y
y
We’ll need to use a substitution for the y integral:
dudyyy2dydu
yu212
22 y21u
21u
21y eeduedyey
Cte 2y21 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
dtt2dyey
dtt2e
dyy
2
2
y
y
We’ll need to use a substitution for the y integral:
dudyyy2dydu
yu212
22 y21u
21u
21y eeduedyey
Cte 2y21 2
Ct2e 2y2
C is an arbitrary constant, so we can multiply by -2 and still call it C
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
dtt2dyey
dtt2e
dyy
2
2
y
y
We’ll need to use a substitution for the y integral:
dudyyy2dydu
yu212
22 y21u
21u
21y eeduedyey
Cte 2y21 2
Ct2e 2y2
C is an arbitrary constant, so we can multiply by -2 and still call it C
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Use separation of variables to find the general solution to the given differential equation.
22 yy et2dtdy
yet2yy
dtt2dyey
dtt2e
dyy
2
2
y
y
)Ct2ln(y
)Ct2ln(y
2
22
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
dt)t2sin(2du)t2cos(u
dt)t2cos()t2sin(
dt)t2tan(
We can use a substitution for this integral
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
dt)t2sin(2du)t2cos(u
dt)t2cos()t2sin(
dt)t2tan(
We can use a substitution for this integral
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t2cos(lnu
dudt
)t2cos()t2sin(
212
1
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
We can use a substitution for this integral
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
)t2cos(lnC)t2cos(ln
21
21
21
Ceey
C)t2cos(lnyln
dt)t2sin(2du)t2cos(u
dt)t2cos()t2sin(
dt)t2tan(
C)t2cos(lnu
dudt
)t2cos()t2sin(
212
1
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
Now we can find the constant from the initial value:
2CCe2
Ce20
)02cos(ln21
We can use a substitution for this integral
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
)t2cos(lnC)t2cos(ln
21
21
21
Ceey
C)t2cos(lnyln
dt)t2sin(2du)t2cos(u
dt)t2cos()t2sin(
dt)t2tan(
C)t2cos(lnu
dudt
)t2cos()t2sin(
212
1
Find a particular solution to the given initial value problem.
2)0(y);t2tan(yy
First we separate and integrate
dt)t2tan(y
dy
)t2tan(ydtdy
Now we can find the constant from the initial value:
2CCe2
Ce20
)02cos(ln21
Plugging this into our general solution we get
)t2cos(
2e2y
)t2cos(lnp
21
We can use a substitution for this integral
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
)t2cos(lnC)t2cos(ln
21
21
21
Ceey
C)t2cos(lnyln
dt)t2sin(2du)t2cos(u
dt)t2cos()t2sin(
dt)t2tan(
C)t2cos(lnu
dudt
)t2cos()t2sin(
212
1
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnyln23
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnyln23
tln23
Cey
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnyln23
tln23
Cey
23
Cty
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnyln23
tln23
Cey
23
Cty
4C)1(C4 23
Find C from the initial value
Find a particular solution to the given initial value problem.
4)1(y;y3yt2
t2dt3
ydy
y3dtdy
t2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnyln23
tln23
Cey
23
Cty
4C)1(C4 23
23
t4yp
Find C from the initial value
Find a particular solution to the given initial value problem.
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
1t
dtdyy
2
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
1t
dtdyy
2
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t(tany 1221
Find a particular solution to the given initial value problem.
1t
dtdyy
2
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t(tan2y 12
C)t(tany 1221
Find a particular solution to the given initial value problem.
1t
dtdyy
2
Don’t forget the +/-
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t(tan2y 12
C)t(tany 1221
C)t(tan2y 1
Find a particular solution to the given initial value problem.
1t
dtdyy
2
Don’t forget the +/-
9CC3
C)0(tan23 1
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t(tan2y 12
C)t(tany 1221
C)t(tan2y 1
Find a particular solution to the given initial value problem.
1t
dtdyy
2
Don’t forget the +/-
9CC3
C)0(tan23 1
9)t(tan2y 1p
3)0(y;1yy)1t( 2
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
C)t(tan2y 12
C)t(tany 1221
C)t(tan2y 1
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnlnyln
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnlnyln
tlnCeyCtlnln
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnlnyln
tlnCeyCtlnln
2lnC)4ln(
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnlnyln
tlnCeyCtlnln
2lnC)4ln(
2)2ln()2ln(2
)2ln()2ln(
)2ln()4ln(
C2
Find a particular solution to the given initial value problem.
)4ln()2(y;yy)tln(t
)tln(tdt
ydy
ydtdy
)tln(t
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB
Ctlnlnyln
tlnCeyCtlnln
2lnC)4ln(
2)2ln()2ln(2
)2ln()2ln(
)2ln()4ln(
C2
tln2yp