# Homogeneous, Particular, and General Solutions

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Y''(t) + 6y'(t) + 13y(t) = te^(-t)

Compute the homogeneous solution yh(t).

Compute the particular solution yp(t).

Compute the general solution y(t) if y(0)=0 and y'(0)=1/8

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#### Solution Preview

Homogeneous Solution:

Aux. equation is: m^2 + 6m +13 = 0

m = (1/2)[-6 (+/-) sqrt(36-52) = -3 (+/-) i2

Two complex roots, m1 = -3+i2 and m2 = -3-i2

Hence, the homo. solution is:

yh(t) = e^(-3t) * [C1*Cos 2t + C2*Sin 2t]

Particular Solution:

Forcing function = te^(-t)

So, solution should ...

#### Solution Summary

The solution explains clearly and concisely the calculations necessary to reach the homogeneous, particular and general solutions of the given equation.

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