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{\bf Question}

Find the general solution of the differential equation: $$4
\frac{d^2x}{dt^2} - 4 \frac{dx}{dt} + x = e^{-t}$$

\vspace{.25in} {\bf Answer}

$$4 \frac{d^2x}{dt^2} - 4 \frac{dx}{dt} + x = e^{-t}$$

Complementary Function:

auxiliary equation $4m^2 - 4m + 1 = 0 \Rightarrow (2m -1)^2 = 0$

TWO solutions with $m = \frac{1}{2}$

Hence the Complementary Function is $x_c=(At+B)e^{\frac{1}{2}t}$

Particular integral method of undetermined coefficients
\begin{eqnarray*} {\rm Let \ \ \ \ } x^* & = & Ce^{-t} \\
4 \frac{d^2x^*}{dt^2} - 4 \frac{dx^*}{dt} + x & = & Ce^{-t}[4 + 4
+ 1] \equiv e^{-t} \\ {\rm Hence\ \ \ \ \ } C & = & \frac{1}{9}
\end{eqnarray*}
Hence the general solution is now $$x = (At + B)e^{\frac{1}{2}t} +
\frac{1}{9}e^{-t}$$



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