Science:Math Exam Resources/Courses/MATH215/December 2011/Question 04 (b)
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Question 04 (b) 

Suppose the motion of a springmass system is described by the following differential equation (b) Solve the following equation for u(t) in explicit form. 
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Hint 

What are the particular and homogeneous solutions? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Notice that the ODE has a nonzero righthand side. We will begin by solving the homogeneous equation By making the ansatz we arrive at with roots . Complex roots of the form lead to solutions of the form Therefore the homogeneous solution is The righthand side is sinusoidal with argument . We need to think of which functions (and their derivatives) that could lead to this possible righthand side. The only such possible functions are and so, we will find a particular solution of the form Taking this particular solution and plugging it in to the ODE, we have: In comparing the coefficients of and , we find that The general solution to the ODE will be , giving The initial conditions tell us From this, we have, and Therefore we get that The solution is thus 