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Mathematics 238 first test
Monday, July 7, 2008
please show your work to get full credit for each problem
1. Is y = e4x a solution to the differential equation y 00 − 5y 0 + 4y = 0 ?
2. Is y =
Z
x
cos t2 dt a solution of the initial value problem
0
3. Given the differential equation
dy
= cos x2 and y(0) = 0 ?
dx
dy
= (x2 sec 3x)y
dx
(a) Is the equation separable?
(b) Is the equation linear?
(c) Is the equation exact?
4. Given the differential equation 3x2 y 2 + 2x3 y + 12y 2
dy
dx
=0
(a) Is the equation separable?
(b) Is the equation linear?
(c) Is the equation exact?
√
dy
3
3
5. Solve the initial value problem
= y (2x−5) and y(0) =
and state the solution’s domain.
dx
6
6. Solve the initial value problem
dy
+3y = 6x and y(0) = 2.
dx
7. Solve the differential equation x
dy
−2y = x3 sin 4x.
dx
8. Solve the initial value problem
9. Solve the differential equation
−2x−3 y 3 +
1
dx+ 3x−2 y 2 + 2e2y dy = 0 and y(1) = 0.
x
dy
√
√
+4y = 2e−2x y using the substitution w = y
dx
page two
10. Sketch the direction field & representative isoclines for
11. Suppose that
dy
x
= 2+
dx
y
dP
= f (P ) where f is sketched below
dt
(a) What are the equilibrium (or constant) solutions?
(b) Sketch a phase line for P
(c) Classify each equilibrium solution as stable, unstable, or semi-stable.
√
1
dy
y = 2005 − x and y(1) = 3
12. For which values of x is a solution of (log(2001 − x)) +
dx
x − 2004
guaranteed to exist?
dy
√
13. Is a solution of the initial value problem
= x2 sin( 3 y) and y(π) = 0
dx
guaranteed to exist & to be unique? Explain.
dy
= 3x−y 3 and y(0) = 2
dx
find some approximate values for y(x) using Euler’s method with step-size h = 1/2.
Complete the following table:
14. Given the differential equation
x
y
0
2
dy
dx
approximate ∆y
0.5
15. A twenty year loan of $ 250,000 is subject to an interest rate of 5%, compounded
continuously, and is repaid continuously at a rate of k dollars per year.
Denote by P (t) the amount owed at time t years.
(a) Write down a differential equation for P (t).
(b) Solve the differential equation, and find the value of k.