# Is there any easy way to solve this differential equation?

I was studying the topic "Nonhomogenous second order linear differential equations: Method of Undetermined Coefficients"

And I am asked to find the general solution of the following differential equation:

$$y''+3y'+2y = e^{t}(t^2+1)\sin(2t) + 3e^{-t}\cos(t)+6e^{t}$$

In finding a particular solution I assume the following function solve the DE and try to find the corresponding coefficients :

\begin{align*} Y & = Y_1+Y_2+Y_3 \text{ where } \\ Y_1 & = (At^2+Bt+C)[\sin(2t)+\cos(2t)]e^t \\ Y_2 & = [A\sin(2t) + B\cos(2t)]e^{-t} \\ Y_3 & = Ae^{t} \end{align*} where $A,B,C$ are the constants them I'm required the find through plugging them into the DE.

My questions:

1 : Is my following setup for $Y_1,Y_2,Y_3$ correct assuming that they are not part of the homogenous solution of the corresponding equation?

2 : Is there any easy way to solve this DE using method of undetermined coefficients? Am I missing a trick or something? Or does the book just want to give me cancer?

Thanks.

• The ODE is linear, so I would actually at least try the Laplace transform first. At the very least, it would clarify what algebraic form I should expect a solution to have. – avs Jul 17 '17 at 17:20
• @Xenidia: Why do you have $Y_2 = e^{-t}(a \sin 2t + b \cos 2t)$? Those should be $\cos t$ and $\sin t$ terms. The approach is sound otherwise. Laplace Transforms or Variation of Parameters are other approaches. – Moo Jul 17 '17 at 17:26
• Do you have any initial conditions that makes terms vanish in a Laplace transform, or are you trying to solve this with full generality? – GFauxPas Jul 17 '17 at 18:24

The auxiliary equation is given by

$$m^2+3m+2=0$$

The complementary solution is

$$y_c=c_1e^{-2t}+c_2e^{-t}$$

Hint the set of undetermined coefficients are

$$S_1= {t^2e^{t}\sin2t,t^2e^{t}\cos2t,te^{t}\sin2t,te^{t}\cos2t+e^{t}\cos2t,e^{t}\sin2t}$$

$$S_2={e^{t}\cos2t,e^{t}\sin2t}$$

$$S_3={e^{-t}\cos t,e^{-t}\sin t}$$

$$S_4={e^t}$$

Since $S_2$ is inside $S_1$

Then we take $S_1$ and ignore $S_2$

The UC set is then

$$S_1= {t^2e^{t}\sin2t,t^2e^{t}\cos2t,te^{t}\sin2t,te^{t}\cos2t+e^{t}\cos2t,e^{t}\sin2t}$$

$$S_3={e^{-t}\cos t,e^{-t}\sin t}$$

$$S_4={e^t}$$

Hope that you can carry on!

There are some mistakes in yours. Eg. $e^{-t}cos2t$