280, Math. Spring 2009 Various practice problems Problem 1. Find values of m so that y = exp(mt) is a solution of (a) dy + 2y = 0, dt (b) d2 y - y = 0, dt2 (c) d3 y d2 y dy -3 2 +2 = 0. 3 dt dt dt
Problem 2. Determine the values of r for which the following differential equation t2 has solutions of the form y = tr for x > 0. Problem 3. Solve the initial value problem dy = y 2 cos(x), dx 1 y(0) = - . 2 dy d2 y + 4t + 2y = 0 2 dt dt
Problem 4. Solve the following differential equations by separation of variables. dy y2 , = dx 1 + x2 + y 2 + x2 y 2 dN + N = N t exp(t + 2), dt
(a)
(b)
(c)
exp(x)
dy = 2x, dx
(d)
y exp(x)
dy = exp(-y) + exp(-2x - y). dx
Problem 5. Verify that the indicated functions are solutions of the given differential equation. dy (a) 2ty + (t2 + 2y) = 0; t2 y + y 2 = c, where c R is arbitrary. dt HINT : Use implicit differentiation. (b) y + 2ty = 1; y = exp(-t2 )
t 2 0 exp(s ) ds
+ c exp(-t2 ), where c R is arbitrary.
Problem 6*. Let a R be a nonzero constant and let b1 , b2 be two continuous functions on 0 x < such that |b1 (x) - b2 (x)| k, for some k > 0. Let be a solution of y + ay = b1 (x), and be a solution of y + ay = b2 (x). Assume that (0) = (0). Show that k [1 - exp(-ax)] , for 0 x < . a HINT : Write down explicit solutions to equations (1) and (2) and take their difference; then evaluate the resulting integral. |(x) - (x)| Problem 7*. Suppose is a function with continuous derivative on [0, b), for some 0 < b , satisfying (x) - 2(x) 1, Show that (x) and (0) = 1. (3) (2) (1) 0 x < ,
1 3 exp(2x) - . 2 2
1
2
HINT :
Multiply (3) by an integrating factor corresponding to the equation y - 2y = g(x) and then integrate.
Problem 8. In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. In addition, the amount of material forgotten is proportional to the amount memorized at time t. Suppose that M denotes the total amount of a subject to be memorized and A(t) is the memorized amount at time t. (a) (b) Determine a differential equations for the amount A(t). Solve the differential equation from part (a) with the initial condition A(0) = A0 > 0.
Problem 9. A thermometer is taken from an inside room to the outside, where the air temperature is 5 F. After 1 minute the thermometer reads 55 F, and after 5 minutes the reading is 30 F. What was the initial temperature of the room ? NOTE. Use Newton's law of cooling. Problem 10*. (a) Find a solution of the initial-value problem y = t uniqueness theorem? Explain. (b) 1 - y 2 , y(0) = -1, other than y(t) = -1. What about 1 - y 2 , y(0) = 1.
Show that y(t) = 1 is the only solution of the initial-value problem y = t
Problem 11. Let y1 (t), y2 (t), and y3 (t) be three distinct (non-zero) solutions of y + a(t)y = b(t). Prove that for all t y2 (t) - y1 (t) = c, y3 (t) - y1 (t) (a, b continuous) for some constant c.
Problem 12. Show that every solution of y + tends to zero as x . Problem 13*. The graph of a certain function y = f (x) passes through the origin. If (x, y) is a point (= (0, 0)) on the graph in the first quadrant, then the graph divides the rectangle with vertices (0, 0), (x, 0), (x, y), and (0, y) into two parts A and...