elementary differential equation
DESCRIPTION
PDF file of the Elementary Differential Equation. Problem and Solutions ONLY that found in MATHalino.comTRANSCRIPT
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Elimination of Arbitrary
Constants
Problem 01
Solution 01
Divide by 3x
answer
Problem 02
Solution 02
answer
Problem 03
Solution 03
→ equation (1)
Divide by dx
Substitute c to equation (1)
Multiply by dx
answer
Another Solution
okay
Problem 04
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Solution 04
→ equation (1)
Substitute c to equation (1)
answer
Another Solution
Divide by y2
Multiply by y3
okay
Separation of Variables | Equations of Order
One
Problem 01
, when ,
Solution 01
HideClick here to show or hide the solution
when ,
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then,
answer
Problem 02
, when , .
Solution 2
when ,
then,
answer
Problem 03
, when , .
Solution 03
when ,
then,
![Page 4: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/4.jpg)
answer
Problem 04
, when , .
Solution 04
Therefore,
when x = 2, y = 1
Thus,
answer
Problem 05
, when , .
Solution 05
From Solution 04,
when x = -2, y = 1
Thus,
answer
Problem 06
, when , .
Solution 06
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From Solution 04,
when x = 2, y = -1
Thus,
answer
Problem 07
, when , .
Solution 07
when x = 0, y = 0
thus,
answer
Problem 08
, when
, .
Solution 08
![Page 6: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/6.jpg)
For
Let
,
,
Then,
when x → ∞, y → ½
Thus,
answer
Problem 09
, when
, .
Solution 09
when θ = 0, r = a
![Page 7: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/7.jpg)
Thus,
answer
Problem 10
, when , .
Solution 10
when x = xo, v = vo
thus,
answer
Problem 11
Solution 11
answer
Problem 12
Solution 12
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answer
Problem 13
Solution 13
answer
Problem 14
Solution 14
answer
Problem 15
Solution 15
answer
Problem 16
Solution 16
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answer
Problem 17
Solution 17
answer
Problem 18
Solution 18
answer
Problem 19
Solution 19
answer
Problem 20
Solution 20
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answer
Problem 21
Solution 21
By long division
Thus,
ans
wer
Problem 22
Solution 22
By long division
Thus,
answer
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Problem 23
Solution 23
answer
Homogeneous Functions | Equations of Order
One
Problem 01
Solution 01
Let
Substitute,
Divide by x2,
From
Thus,
answer
Problem 02
Solution 02
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Let
Substitute,
From
answer
Problem 03
Solution 03
Let
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From
Thus,
answer
Problem 04
Solution 04
Let
From
Thus,
answer
Exact Equations | Equations of Order One
Problem 01
Solution 01
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Test for exactness
;
;
; thus, exact!
Step 1: Let
Step 2: Integrate partially with respect to x,
holding y as constant
→ Equation (1)
Step 3: Differentiate Equation (1) partially with
respect to y, holding x as constant
Step 4: Equate the result of Step 3 to N and
collect similar terms. Let
Step 5: Integrate partially the result in Step 4
with respect to y, holding x as constant
Step 6: Substitute f(y) to Equation (1)
Equate F to ½c
answer
Problem 02
Solution 02
Test for exactness
Exact!
Let
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Integrate partially in x, holding y as constant
→ Equation (1)
Differentiate partially in y, holding x as constant
Let
Integrate partially in y, holding x as constant
Substitute f(y) to Equation (1)
Equate F to c
answer
Problem 03
Solution 03
Test for exactness
Exact!
Let
Integrate partially in x, holding y as constant
→ Equation (1)
Differentiate partially in y, holding x as constant
Let
![Page 16: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/16.jpg)
Integrate partially in y, holding x as constant
Substitute f(y) to Equation (1)
Equate F to c
answer
Problem 04
Solution 04
Test for exactness
Exact!
Let
Integrate partially in x, holding y as constant
→ Equation
(1)
Differentiate partially in y, holding x as constant
Let
Integrate partially in y, holding x as constant
Substitute f(y) to Equation (1)
Equate F to c
answer
Linear Equations of Order One
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Problem 01
Solution 01
→ linear in y
Hence,
Integrating factor:
Thus,
Multiply by 2x3
answer
Problem 02
Solution 02
→ linear in
y
Hence,
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Integrating factor:
Thus,
Mulitply by (x + 2)-4
answer
Problem 03
Solution 03
→ linear in y
Hence,
Integrating factor:
Thus,
Using integration by parts
,
,
Multiply by 4e-2x
answer
Problem 04
Solution 04
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→ linear in x
Hence,
Integrating factor:
Thus,
Using integration by parts
,
,
Multiply 20(y + 1)-4
answer
Integrating Factors Found by Inspection
Problem 01
Solution 01
Divide by y2
Multiply by y
answer
Problem 02
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Solution 02
Divide by y3
answer
Problem 03
Solution 03
Divide by x both sides
answer
Problem 04
Solution 04
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Multiply by s2t2
answer
Problem 05
Problem 05
answer
- See more at:
Problem 06
Solution 06
answer
Problem 07
Solution 07 - Another Solution for Problem 06
Divide by xy(y2 + 1)
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Resolve into partial fraction
Set y = 0, A = -1
Equate coefficients of y2
1 = A + B
1 = -1 + B
B = 2
Equate coefficients of y
0 = 0 + C
C = 0
Hence,
Thus,
answer - okay
Problem 11
Solution 11
answer
The Determination of Integrating Factor
Problem 01
Solution 01
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→ a function of
x alone
Integrating factor
Thus,
answer
Problem 02
Solution 02
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→
a function of x alone
Integrating factor
Thus,
answer
Problem 03
Solution 03
→
neither a function of x alone nor y alone
→ a function of y alone
Integrating factor
Thus,
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answer
Problem 04
Solution 04
→
neither a function of x alone nor y alone
→
a function y alone
Integrating factor
Thus,
![Page 26: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/26.jpg)
answer
Substitution Suggested by the Equation |
Bernoulli's Equation
Problem 01
Solution 01
Let
Thus,
→ vari
ables separable
Divide by ½(5z + 11)
ans
Problem 02
Solution 02
Let
Hence,
→ homogeneo
us equation
Let
![Page 27: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/27.jpg)
Divide by vx3(3 + v)
Consider
Set v = 0, A = 2/3
Set v = -3, B = -2/3
Thus,
From
But
answer
Problem 03
![Page 28: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/28.jpg)
Solution 03
Let
But
answer
Problem 04
Solution 04
→ Bernoulli's
equation
From which
Integrating factor,
Thus,
![Page 29: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/29.jpg)
answer
Problem 05
Solution 05
Let
answer
Elementary Applications
Newton's Law of Cooling
Problem 01
A thermometer which has been at the reading
of 70°F inside a house is placed outside where
the air temperature is 10°F. Three minutes later
it is found that the thermometer reading is
25°F. Find the thermometer reading after 6
minutes.
Solution 01
According to Newton’s Law of cooling, the
time rate of change of temperature is
proportional to the temperature difference.
When t = 0, T = 70°F
![Page 30: Elementary differential equation](https://reader034.vdocuments.site/reader034/viewer/2022042501/55766032d8b42a8f138b4590/html5/thumbnails/30.jpg)
Hence,
When t = 3 min, T = 25°F
Thus,
After 6 minutes, t = 6
answer
Simple Chemical Conversion
Problem 01
Radium decomposes at a rate proportional to
the quantity of radium present. Suppose it is
found that in 25 years approximately 1.1% of
certain quantity of radium has decomposed.
Determine how long (in years) it will take for
one-half of the original amount of radium to
decompose.
Solution 01
When t = 25 yrs., x = (100% - 1.1%)xo = 0.989xo
Thus,
When x = 0.5xo
answer
Problem 02
A certain radioactive substance has a half-life of
38 hour. Find how long it takes for 90% of the
radioactivity to be dissipated.
Solution 02
When t = 38 hr, x = 0.5xo
Hence,
When 90% are dissipated, x = 0.1xo
answer