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

rocedwrp 2021-01-02 Answered

Solve the Differential equations
(d2y)dt2+4dydt+3y=et

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Expert Answer

Layton
Answered 2021-01-03 Author has 89 answers

Given IVP is y+4y+3y=et. To find the auxiliary equation, let y=em×x. The auxiliary equation of is
(m2)+4m+3=0
(m+3)(m+1)=0
m=3,1
So, the complimentary function is given by
ycf=c1(e3x)+c2ex.

The particular integral is given by

yπ=1/((D2)+4d+3)et

=et((1/(D1)2)+4(D1)+3)1

=et(1/(D2)2D+1+4D4+3)1

=et((1/2D)(D/2)+1)11

=et(1/2D)1

=t(et)/2

Therefore the solition is
y=ycf+yπ=c1(e3x)+c2(ex)+tet2

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