18.034 MIDTERM 2

Explain your answers clearly; show all steps. Calculators may not be used. All problems have equal value. Please put your name on every sheet. Good luck!

1. (a) y_{1}, y_{2}, and y_{3} are 3 solutions of the differential equation (1 − t)y′′′ + y′′ + t^{2}y′ + t^{3}y = 0 on the interval 1 < t < ∞. Calculate the function W (y_{1}, y_{2}, y_{3})(t) given that W (y_{1}, y_{2}, y_{3})(2) = 3.

(b) The equation y′ + a(x)y = 0 has for a solution

ϕ(x) = e^{− ∫_{x_{0}}^{x} a(t)dt}.

(Here let a be continuous on an interval I containing x_{0}.) This suggests trying to find a solution of

L(y) = y′′ + a_{1}(x)y′ + a_{2}(x)y = 0

of the form

ϕ(x) = e^{∫_{x_{0}}^{x} p(t)dt}

where p is a function to be determined. Show that ϕ is a solution of L(y) = 0 if and only if p satisfies the first-order non-linear equation y′ = −y^{2} − a_{1}(x)y − a_{2}(x). (Remark: This last equation is called a Riccati equation.)

2. (a) Consider the equation y′′ − \frac{2}{x^{2}} y = 0 (for 0 < x < ∞). Find all solutions. (Hint: Try functions of the form y = x^{r}. How do you know you’ve found all the solutions?)

(b) Find all solutions to the equation y′′ − \frac{2}{x^{2}} y = x. Hint: Use “variation of parameters”. Suppose ϕ_{1} and ϕ_{2} are linearly independent solutions to the homogeneous version of the equation (see (a)). Look for a solution of the form ϕ(x) = u_{1}(x)ϕ_{1}(x) + u_{2}(x)ϕ_{2}(x).

3. Iterate x → \sqrt{1 + x}. Start with x = 0. What happens?

4. (a) State the Existence and Uniqueness Theorem for differential equations of the form y′ = f(x, y).

(b) Consider the differential equation y′ = t^{2}(y + 1) on the interval ℝ, with initial condition y(0) = 0. Find a solution y = ϕ(t) defined for all t ∈ ℝ. If the first few Picard iterates (used in the proof of the Existence and Uniqueness Theorem described in (a)) are ϕ_{0}(t) = 0, ϕ_{1}(t), ϕ_{2}(t), find ϕ_{1}(t) and ϕ_{2}(t).

(c) Explain why the ϕ_{1}(t) and ϕ_{2}(t) you found are approximations to ϕ(t).

Date: April 14, 2000, 1:05-1:55 pm.

5. Consider the equation y′′ + cos(x)y′ + sin(x)y = 0.

(a) Let ϕ(x) be a nontrivial solution, and let ψ(x) = ϕ(x + 2π). Prove that ψ(x) is also a solution.

(b) Show that ϕ(x) is a periodic solution of period 2π if, and only if, ϕ(0) = ϕ(2π) and ϕ′(0) = ϕ′(2π).

(c) Let ϕ_{1}(x), ϕ_{2}(x) be two solutions satisfying ϕ_{1}(0) = 1, ϕ′_{1}(0) = 0, ϕ_{2}(0) = 0, ϕ′_{2}(0) = 1. Show that there are constants a and b such that

ϕ_{1}(x + 2π) = aϕ_{1}(x) + bϕ_{2}(x).

(Hint: See (a).)

6. Let A = ( \begin{matrix}3 & 1 \\ −5 & −3 \end{matrix} ). Find the eigenvalues of A. Find eigenvectors of A corresponding to each of the eigenvalues. Calculate A^{2000}.