Answer to Question #137969 in Differential Equations for shan

Question

Solve the following system of ODE:

x' - y - y' = -e^t

Answer 1

$x' - y - y' = -e^t\\ \displaystyle\frac{\mathrm{d}x}{\mathrm{d}t} - y - \frac{\mathrm{d}y}{\mathrm{d}t} = -e^t\\ \textsf{If}\hspace{0.1cm} y = x = e^{mt}, \frac{\mathrm{d}y}{\mathrm{d}t} = \frac{\mathrm{d}x}{\mathrm{d}t} = me^{mt}\\ \implies me^{mt} - e^{mt} - me^{mt}= -e^t\\ -e^{mt} = -e^{t} \implies m = 1\\ m = 1\hspace{0.1cm}\textsf{is the only solution}\\\textsf{to the equation.}\\ \therefore y = x = e^{t} \hspace{0.1cm}\textsf{is a solution to the first}\\\textsf{order linear differential equation.}$