DIFFERENTIAL EQUATIONS
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DIFFERENTIAL EQUATIONS
EXERCISE SET
First order ordinary differential equations.
1. Find the general solution of the following separable differential equations:
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
2. Solve the following initial value problems:
a) [pic], [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
3. Using the transformation y/x=u to find the solution of
a) [pic] b) [pic]
4. Using the indicated transformation, find the general solution:
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
5. Solve the following linear differential equations:
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
6. Solve the following initial value problems:
a) [pic] b) [pic]
c) [pic] d) [pic]
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7. Using the indicated transformation, find the general solution:
a) [pic] b) [pic]
c) [pic] d) [pic]
Higher order ordinary differential equations.
1. Find the general solution of the following differential equations.
a) [pic] b) [pic]
c) [pic] d) [pic]
2. Solve the following initial value problems.
a) [pic] b)[pic]
c) [pic] d) [pic]
3. Find the general solution of the following differential equations.
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
4. Solve the following initial value problems.
a) [pic]
b) [pic]
c) [pic]
d) [pic]
5. Find the general solution of the following differential equations.
a) [pic] b) [pic]
c) [pic] d) [pic]
6. Find the general solution ( use the transformation [pic] ).
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
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7. Solve the following differential equations.
a) [pic] b) [pic]
8. Find the general solution of the following differential equations.
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
9. Solve the following system of differential equations.
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h)[pic]
i) [pic] j) [pic]
k) [pic] l) [pic].
Exact Differential Equations
1. Show that the differential equation is exact and solve it
a) [pic] b) [pic]
c) [pic] d) [pic]
2. Find an integrating factor and solve it
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
3. Give the particular solution satisfying the initial condition
a) [pic], y(1)=0 b) [pic], y(0)=2
- 4 -
Use Picard′s method to obtain n successive approximations of the solution
1. [pic], y(0)=1, n=4
2. [pic], y(0)=0, n=3
3. [pic], y(0)=0, n=4
4. [pic], y(0)=0, n=3
Use the power series (Taylor series) to solve the initial-value problems
1. [pic], y(0)=0
2. [pic], y(0)=0
3. [pic], y(0)=1
4. [pic], y(0)=y′(0)=1
5. [pic], y(0)=0, y′(0)=1
6. [pic], y(0)=1, y′(0)=0
Answers
First order differential equations
1. a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic], [pic] g) [pic] h) [pic] i) [pic]
j) [pic]
2.a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
3.a) [pic] b) [pic]
4.a) [pic] b) [pic], [pic] c) [pic]
d) [pic] e) [pic]
f) [pic]
5.a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
6.a) [pic] b) [pic] c) [pic] d) [pic]
7.a) [pic] b) [pic]
- 5 -
c) [pic] d) [pic]
Higher order differential equations
1.a) [pic] b) [pic]
c) [pic] d) [pic]
2.a) [pic] b) [pic] c) [pic]
d) [pic]
3.a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
i) [pic] j) [pic]
4.a) [pic] b) [pic]
c) [pic] d) [pic]
5.a) [pic] b) [pic]
c) [pic] d) [pic]
6.a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
7.a) [pic] b) [pic]
8.a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic] f) [pic]
g) [pic] h) [pic]
9.a) [pic] b) [pic] c) [pic]
d) [pic] e) [pic]
f) [pic] g) [pic]
- 6 –
h) [pic] i) [pic]
j) [pic] k) [pic]
l) [pic]
Exact differential equations
1.a) [pic] b) [pic]
c) [pic] d) [pic]
2.a) m(x)=1/x, [pic] b) [pic], [pic]
c) [pic], [pic] d) m(x)=cosx, [pic]
e) [pic], [pic] f) [pic], [pic]
g) [pic], [pic] h) [pic], [pic]
3.a) [pic] b) [pic]
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