Fall 2001  Math 2090  Exam 1

1. (a) Solve the initial value problem \frac{dy}{dx} = \frac{x(2y − 1)}{x^{2} + 1}, y(0) = 1.

y = (x^{2} + 1)/2 solves the IVP

(b) Briefly explain why this initial value problem has a unique solution.

Use the existence and uniqueness theorem for first-order IVPs.

2. Find the general solution of each of the following differential equations.

(Your solutions should be real-valued.)

(a) y′ + \frac{2}{x} y = 8x  y = 2x^{2} + Cx^{−2}
(b) y′′ + 6y′ + 10y = 0  y = c_{1}e^{−3x} cos x + c_{2}e^{−3x} sin x
(c) y′′ − 4y′ + 4y = 4x  y = c_{1}e^{2x} + c_{2}xe^{2x} + x + 1

3. (a) Solve the initial value problem y′′ + 2y′ − 3y = 0, y(0) = 5, y′(0) = 1.

y = e^{−3x} + 4e^{x}

(b) Find a particular solution of the differential equation y′′ + 2y′ − 3y = 8e^{x}.

y = 2xe^{x}

4. Consider the differential equation x^{2}y′′ − xy′ + y = 0, x > 0.

(a) Verify that the function y_{1}(x) = x ln(x) is a solution of this differential equation. If y_{1} = x ln(x), then y′_{1} = ln(x) + 1 and y′′_{1} = x^{−1}. So x^{2}y′′_{1} − xy′_{1} + y_{1} = x^{2}x^{−1} − x(ln(x) + 1) + x ln(x) = 0.

(b) Find all values of r for which the function y(x) = x^{r} is a solution of this differential

equation.

If y = x^{r}, then y′ = rx^{r−1} and y′′ = r(r − 1)x^{r−2}. So x^{2}y′′ − xy′ + y = r(r − 1)x^{r} − rx^{r} + x^{r} = (r^{2} − 2r + 1)x^{r}. So y = x^{r} is a solution if r = 1.

(c) Find the general solution of this differential equation on the interval I = (0, ∞).

Give a thorough explanation why the solution you give is the general solution. The general solution is y = c_{1}x ln(x) + c_{2}x. The solutions y_{1} = x ln(x) and y_{2} = x not proportional on I, so are linearly independent on I.

This can also be established using the Wronskian: W [x ln(x), x] = −x ≠ 0 for x > 0

5. A certain antique coin increases in value at a rate proportional to the square root of the current value. The value of the coin was $100 in the year 2000. Write down the initial value problem you would solve to determine the value of the coin t years from 2000.

If V (t) is the value of the coin t years from 2000, \frac{dV}{dt} = k\sqrt{V} , V (0) = 100

Extra Credit. If the value of the coin in problem 5 will be $144 next year (in 2002), when will the value of the coin be $400? In the year 2010.