# Solve left(d^{2}frac{y}{dt^{2}}right) + 7left(frac{dy}{dt}right) + 10y=4te^{-3}t with y(0)=0, y'(0)= -1

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Benedict

Given:

with
The auxiliary equation is given by
solving this we get
Hence the complimentary function is
Now The P.I. of given differential equation is

Hence the general solution is $y={y}_{c}+{y}_{p}={C}_{1}{v}^{-2}t+{C}_{2}{e}^{5}t-{e}^{-3}t\left(2t+1\right)$
Now (1)
Also
Hence (2)
solving (1) and (2) we get
Hence the solution is

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