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name: 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.