MATH 361 FINAL EXAM FALL 2001

Name:

1. Solve the initial value problem:

t^{3}y′ + 4t^{2}y = e^{−t}; y(−1) = 0.

Ans: .

2. Solve the differential equation (e^{x} sin y − 2y sin x − 2x)dx + (e^{x} cos y + 2 cos x + 3y)dy = 0,

Ans: .

3. A tank contains 120 liters of water. A mixture of 2 g/liter of salt water enters the tank at a rate of 3 liters/minute. The well stirred solution leaves at the same rate. Find the amount of salt in the tank after 12 minutes.

Ans: .

Name:

4. Graph qualitatively the solutions of \frac{dx}{dt} = 12x − 3x^{3} for x(0) = −1, 1, 3. Determine the inflection points and classify the equilibrium solutions.

5. a) Solve the equation: 16y′′ + 24y′ + 9y = 0

Ans: .

b) Solve the equation: y^{iv} + 2y′′ + y = 0

Ans: .

6. a) Solve: y′′ + 2y′ + y = 2e^{−t}

Ans: .

b) Solve: y′′ + 4y = 4 cos 2t, y(0) = 0, y′(0) = 5

Ans: .

Name:

7. Given the Euler equation x^{2}y′′ − 3xy′ + 4y = 0:

a) Find a solution of the form y = x^{p}.

b) Use reduction of order to find the general solution

Ans: .

8. a) Find the general solution of x^{2}y′′ − 3xy′ + 4y = x^{2} ln x. (See pb. (7).)

Ans: .

9. a) Write in the form a + ib: x^{2+4i}

Ans: .

b) Find the radius of convergence of the series solution of (1 + x^{2})y′′ − xy′ + y = 0, about x = 1/2.

Ans: .

(Extra space)

Name:

10. a) Use infinite series to find a solution of the Chebyshev equation (1 − x^{2})y′′ − xy′ + α^{2}y = 0. b) Find polynomial solutions for α = 0, 2.

a) .

b) .

(Extra Space)