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use variation parameters to find a general solution to the differential equation given that the function y1 and y2 are linearly independent solution

ty''-(t+1)y'+y=t^2; y1=e^t,y2=t+1

Verified Answer :

https://youtu.be/X7pzGyfRx9A

1) divide differential equation by t
\(\Large {y''- \frac{(t+1)}{t} y'+\frac1ty=t}\)

2)Solve this system of equations for \(v_1'\) and \(v_2'\)
\(v_1'\left(e^t\right) +v_2'\left(t+1\right)=0 \\ v_1'\left(e^t\right) +v_2'\left(1\right)\ \ \ \ \ =t\)

you get \({v_1}'=\left(t+1\right)/e^t, \ \ {v_2}'=-1\)

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