Solve the Differential equations frac{(d^{2})y)}{d(t^{2}}+4(frac{dy}{dt}+3y=e^{-t}

Solve the Differential equations frac{(d^{2})y)}{d(t^{2}}+4(frac{dy}{dt}+3y=e^{-t}

Differential equations
asked 2021-01-02
Solve the Differential equations

Answers (1)

Given IVP is \(\displaystyle{y}{''}+{4}{y}'+{3}{y}={e}^{{-{t}}}.\) To find the auxiliary equation, let \(\displaystyle{y}={e}^{{{m}}}\times{x}.\) The auxiliary equation of is
So, the complimentary function is given by
The particular integral is given by
PSKypi=1/((D^2)+4d+3)e^-t =e^-t((1/(D-1)^2)+4(D-1)+3)*1 =e^-t(1/(D^2)-2D+1+4D-4+3)*1 =e^-t((1/2D)(D/2)+1)*^-1*1 =e^-t((1/2D)(1-(D/2)+...)*^-1*1 =e^-t(1/2D)*1 =t(e^-t)/2ZSK
Therefore the solition is

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