Please help with this problem with as many steps and as much notation as possibl

Question

Please help with this problem with as many steps and as much notation as possible! Need all parts a-e.

Consider the ODE: ty" + (1 - 2t)y' + (t - 1)y = 1/t et. Verify that y1(t) = et is one solution to the homogeneous problem. Use reduction of order to find another, independent solution to the homogeneous problem. Use variation of parameters to find a solution to the nonhomogeneous problem. Write the general solution to the nonhomogeneous problem. Find the specific solution to the problem with initial conditions y(0) = 2, y' (0) = 1.

Explanation / Answer

a.y"=e^t

y'=e^t

now substituting the values in the equation we get

te^t+(1-2t)*e^t+(t-1)*e^t=(1/t)*e^t

hence proved

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