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

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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]

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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]

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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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