https://youtu.be/X7pzGyfRx9A
1) divide differential equation by t
[tex]\Large {y''- \frac{(t+1)}{t} y'+\frac1ty=t}[/tex]
2)Solve this system of equations for [tex]v_1'[/tex] and [tex]v_2'[/tex]
[tex]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[/tex]
you get [tex]{v_1}'=\left(t+1\right)/e^t, \ \ {v_2}'=-1[/tex]
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