Sample Problems for the Midterm

• The exam is on Tuesday, 02/11, 2:10 pm – 3:30 pm.

1. Find the solutions of the given problem:

(1) y′ − 2y = 0.

(2) y′ = \frac{xy}{x^{2}−1}.

(3) 3y′ + 2xy^{2} = 0, y(0) = 1.

(4) xyy′ = (y^{2} − 1)^{2}.

2. Solve the following first order linear equations:

(1) y′ − 2y = 0.

(2) xy′ + y − 2x = 0.

(3) y′ sin x + y cos x = 1.

(4) y′ + 2xy = x, y(0) = 1.

3. Solve the following equations whenever it is exact:

(1) (x^{2} + y^{2})dx + 2xydy = 0.

(2) (3x + 2y^{2})dx + 2xydy = 0.

(3) (x^{2} + 2xy)dx + x^{2}dy = 0.

(4) ydx + xy^{2}dy = 0.

4. Solve the following Bernoulli equations:

(1) y′ + y = xy^{2}.

(2) 2yy′ sin x + y^{2} cos x = 1, y(π/2) = 0.

(3) y′ + y = y^{2}e^{x}.

(4) y′ + \frac{2y}{x} = −x^{9}y^{5}, y(−1) = 2.

5. Solve following homogeneous equations:

(1) y′ = \frac{x^{2}+2xy+y^{2}}{x^{2}} .

(2) y′ = \frac{y−x}{x}.

(3) y′ = \frac{x^{2}+y^{2}}{2xy}.

6. Use integrating factor method to solve following equations:

(1) (3x + 2y^{2})dx + 2xydy = 0, y(0) = 1.

(2) (y + 1)dx − xdy = 0.

(3) ydx + 3xdy = 0.

(4) 3x^{2}y^{2}dx + (2x^{3}y + x^{3}y^{4})dy.

7. Assume a radioactive material has 16 grams in 2000 and has 1 gram now, what is the half life of it? Give the amount of this material as a function of time t of years. What amount of the material will be still left 10 years later?