Linear differential equation

Question

t(dx/dt)-4x=t^6e^t, x(1)=0

Answer 1

Divide by t: dx/dt - (4/t)x = t⁵e^t

Multiply both sides by the integrating factor e^∫(-4/t)dt = t^-4

t^-4(dx/dt) - 4t^-5x = te^t

We have: dx/dt(xt^-4) = te^t

Integrate both sides with respect to t (use integration by parts for the right side):

xt^-4 = te^t - e^t + C

So, x = t⁵e^t - t⁴e^t + Ct⁴

Since x(1) = 0, e - e + C = 0. So, C = 0

Solution: x = t⁵e^t - t⁴e^t

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