## Differential Equations Practice Exams

Problem 1: Use variation of parameters to find the general solution to First we need to solve the homogeneous equation y''' - y' = 0. Its characteristic equation is . It is easy to see that its root are r=0,1,-1. Therefore we have .
Second we need to find a particular solution using the variation of parameters technique. We have , where u', v', and w' are solution to the system Easy calculations give Integration by parts, will give (you are encouraged to do it) Hence we have which implies that once u, v, w are plugged into the formula giving . Therefore the general solution is given by: Problem 2. Find the solution to the initial value problem where Since g(t) is a step function, we need to use Laplace Transform to solve this problem. Once we attack the equation by , we get which implies . Using the initial condition, we get After easy calculations, we obtain Since , we deduce Hence In order to find y, we need to use the inverse Laplace transform. First we have which implies The hard part concerns the second term . First let us find the inverse Laplace without the exponential. First we know that Using the derivative formula, we get Therefore, we have Using the formula , we get Therefore, we have Problem 3. Find the Laplace transform of We have Hence . Using the formula , we get But which implies [Next Exam] [Calculus] [CyberExam] S.O.S MATH: Home Page

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