Sample Problems for the Midterm

• The exam is on Friday, 02/19, 3:10 pm – 4:00 pm.

• In each problem, you have to show every step of your calculation.

Basic notions:

1. Check that each given function is a solution of the given problem or not: (1) Function: y = 3e^{2x}; Differential equation: y′ − 2y = 0.

(2) Function: y = y(x) satisfying the algebraic equation x^{2} + cy = 0, where c is a constant; Differential equation: y′ = \frac{xy}{x^{2}−1}.

(3) Function: y = e^{−x} − e^{−2x}; Initial value problem: y′′ + 3y′ + 2y = 0, y(0) = 0, y′(0) = 1.

First order equations:

2. Solve the following first order equations:

(1) xy′ + y − 2x = 0.

(2) y′ sin x + y cos x = 1.

(3) xyy′ = (y^{2} − 1)^{2}.

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

(5) (3x + 2y^{2})dx + 2xydy = 0.

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

(7) y′ + y = xy^{2}.

3. Solve the following initial value problems:

(1) y′ + 2xy = x, y(0) = 1.

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

Application problems:

4. An initial deposit of $1,000,000 in a bank with 6% annual interest rate compounded continuously will approximately last how long if it is subject to annual withdrawals of $100,000?

Approximation solutions:

5. Use Euler’s method to compute the approximation solution of the initial value problem

y′ = y + x, y(0) = 1

at x_{1} = 0.1 and x_{2} = 0.2. Compare your approximation with the actual values.