MATH 316, DIFFERENTIAL EQUATIONS, WINTER 2000

Problem Set 1, due Wednesday, January 19

Problem 1 For each of the given (systems of) differential equations, determine what is its order and whether it is linear.

(a) \frac{d^{2}y}{dx^{2}} + (\frac{dy}{dx})^{3} = x.  (b) \frac{d^{3}y}{dx^{3}} + x^{2} \frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} = cos x.

(c) \{ \begin{matrix}\frac{d^{2}y}{dt^{2}} + t^{2}z = 0 \\ \frac{dz}{dt} + t\frac{dy}{dt} = 1. \end{matrix}  (d) \frac{d^{2}y}{dx^{2}} + sin(\frac{dy}{dx}) = y.

(e) (\frac{dy}{dx})^{2} − 3xy \frac{dy}{dx} + x^{2}y^{2} = 0.

Problem 2 Determine the values of r for which the given differential equation has solutions of the form y = e^{rt}.

\begin{matrix}(a) & y′ + 2y = 0 & (b) & y′′ − 2y′ + y = 0 \\ (c) & y′′′ + y′′ − 2y′ = 0 \end{matrix}

Problem 3 (a) Find all solutions to the differential equation \frac{dy}{dt} + 3t^{2}y = t^{2}. (b) Draw a sketch of the slope field and of the integral curves. Is there an equilibrium solution? If yes, is it stable or unstable? (c) Find the solution for which y(0) = 1.

Problem 4 (a) Draw a direction field for the differential equation y′ + y = 2te^{−t} + 3. (b) Based on an inspection of the direction field, describe how solutions behave for large t. (c) Find the general solution and use it to determine how solutions behave as t → ∞.

Problem 5 Consider the initial value problem

3y′ − y = e^{t/5}, y(0) = a.

(a) Draw a direction field. How do solutions appear to behave as t becomes large? Does the behavior depend on the choice of the initial value a? Let a_{0} be the value of a for which the transition from one type of behavior to another occurs. Estimate the value of a_{0}.

(b) Solve the initial value problem and find the critical value a_{0} exactly.

(c) Describe the behavor of the solution corresponding to the initial value a_{0}.

Problem 6 Find the solution of

\frac{dy}{dt} = \frac{1}{e^{y} − t}, y(1) = 0.

Hint: Consider t as the dependent variable instead of y.

Problem 7 Do Problems 2.2 #15, 16 from the textbook.