# Section 3.5 – Nonhomogeneous Equations; Method of Undetermined Coefficients

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it.
1. Find the general solution of the equation/solve the initial value problem
1. $$y''+6y'+9y=t\cos(2t)$$
2. $$4y''+y'=4t^3+48t^2+1$$
3. $$y''+2y'+y=4e^{-t}$$, $$y(0)=2$$, $$y'(0)=1$$​

1. $$y=c_1e^{-3t}+c_2te^{-3t}+\left(\dfrac{5t}{169}+\dfrac{18}{2197}\right)\cos(2t)+\left(\dfrac{12t}{169}-\dfrac{92}{2197}\right)\sin(2t)$$
2. $$y=c_1+c_2e^{-\frac{t}{4}}+t^4+t$$
3. $$y=\left(2+3t+2t^2\right)e^{-t}$$

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2. Find the form of a particular solution for each of the following nonhomogeneous equations.
1. $$y''+2y'+2y=e^{-t}\sin t+e^{-t}\cos 2t$$
2. $$y''-2y'+y=te^{t}+t^2e^{-t}+e^t\cos t+t^2$$​

1. $$y_p=Y=te^{-t}[A\cos(t)+B\sin(t)]+e^{-t}[C\cos(2t)+D\sin(2t)]$$
2. $$y_p=Y=t^2e^t(At+B)+e^{-t}(Ct^2+Dt+E)+e^t(F\cos(t)+G\sin(t))+Ht^2+It+J$$

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