Daada

PRACTICE PROBLEMS

1. First order differential equations

1.1. Integrating factors. This is section 2.1.

(1) Find the general solution to

2

cos(x)

.

y + y=

x

x2

(2) Find the general solution to

y + 2y = xex .

1.2. Separable equations. This is section 2.2.

(1) Solve

y =

x + ex

.

2−y

(2) Find the general solution to

y =

x + 2y

.

2x − y

1.3. Exact equations. This is section 2.6.

(1) Find the general solution to

(x2 + sin(x)ey )y + 2xy + cos(x)ey = 0.

(2) Solve

2yxy + ln(x) + 1 + y 2 = 0.

1.4. Euler’s method. This is section 2.7.

Find the solution to the initial value problem

y = 2y + 1,

y(0) = 1.

Using Euler’s method, write down the recurrence relation with step h

(i.e. write yk+1 as a function of yk ). Solve the recurrence relation for

the general term yk . Show that as h → 0 the approximate solution

converges to the exact solution.

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PRACTICE PROBLEMS

1.5. Picard’s method. This is section 2.8.

Find the exact solution to

y = 1 + y2,

y(0) = 0.

Use Picard’s approximation method to find the first three approximate

solutions φ1 (x), φ2 (x), φ3 (x). Compare them to the Taylor expansion

of the exact solution around x = 0.

2. Second order linear equations

2.1. Equations with constant coefficients. These problems cover

sections 3.1, 3.3, 3.4.

(1) Find the general solution to

y − 4y + 3y = 0.

(2) Find the general solution to

y − 4y + 4y = 0.

(3) Find the general solution to

y + 4y = 0.

2.2. Reduction of order. This is section 3.4.

(1) y = x3 is a solution to

x2 y − 3xy + 3y = 0.

Find the other independent solution.

(2) y = sin(x2 ) is a solution to

xy − y + 4x3 y = 0.

Find the other independent solution.

3. Higher order linear equations

Find the Wronskian (defined up to a constant) of a fundamental set

of solutions to

xy + x sin(x)y − 3y = 0.

PRACTICE PROBLEMS

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3.1. Equations with constant coefficients. This is section 4.2.

(1) Find the general solution to

y (4) − 5y + 4y = 0.

(2)...