Question

Solve the following system of ODE

x'+x-y = e^2t

Answer 1

Given Equation,

$x'+x-y=e^{2t}$

$\frac{dx}{dt}+x=y+e^{2t}$

Integrated factor, i.f=$e^{\int 1.dt}$

=$e^t$

solution is given by,

$x\times i.f.=\int i.f.\times (y+e^{2t})dt+c$

$xe^t=\int(y+e^{2t})e^tdt+c$

$xe^t=ye^t+\frac{e^{3t}}{3}+c$

$x=y+\frac{e^{2t}}{3}+ce^{-t}$