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MATH 239 DIFFERENTIAL EQUATIONS FALL 2008 TEST 1 NAME_______________________________ Show all your work and please box your final answer. 1. Which of the following functions is a solution to y' '2 y 3 0 ? (Show work to support your answers.) y1 e 2t y2 sin( t ) y3 1 t For the solution you just found, is a constant multiple of that solution also a solution? 2. Which of the following functions is a solution to y ' '2 y '8 y 0 ? (Show work to support your answers.) y1 e 2t y2 sin( t ) y3 1 t For the solution you just found, is a constant multiple of that solution also a solution? There's an important difference between the differential equations in 1 and 2. What is that difference? 3. Make up an interesting exact differential equation and prove that it is exact. 4. Find and classify all equilibrium solutions of y y 2 ( y 4) 2 ( y 6) and make a sketch of possible solutions (direction field). Based on your sketch, determine the behavior of y as t and if this behavior depends on the initial value of y at t 0 , describe this dependency. 5. Consider the first order linear differential equation y y t 2 . a) Create a slope field for this diff eq and based on the slope field, explain the end behavior of solutions and/or how the end behavior depends on the choice of initial conditions. b) Now actually solve the diff eq and confirm your conclusion about end behavior by calculating lim t y . 6. Solve the following differential equations, incorporating any given initial conditions. There's one here of each type we know how to solve, trust me. a) dy cos( x) y 8 cos( x), dx y (0) 10 y b) 6 x dx (ln( x) 2)dy 0 x c) 2 sin( y ) cos( x)dx cos( y ) sin( x)dy 0 7. Take a look at this differential equation: ( y y )t y ' for t 0 . Its second order, so as it is we can't solve it using the methods we've seen so far. But the substitution v y and v' y will make it first order in v . a) Do this substitution and solve the first order differential equation to get a general solution for v . b) Now refer back to your original substitution to find a general solution for y . : : Bonus : : If a function is a quadratic, that implies it’s a polynomial. If something's a square, that implies that it's a rectangle. Do any of the types we focused on in Chapter 2 (linear, exact, separable) imply any other type? Provide some justification for your answer.