# Solving Ordinary Nonhomogenous Differential Equations Using Power Series

Edited: Apr 1

Hey Bprp could you make a video on how to solve an ordinary Nonhomogenous differential equation using power series I’ve been trying to prove that it can be done with the ODE y”+2y’+y=e^t because I already know that the answer is c1e^-t +c2te^-t +e^t/4 I just want a clear explanaion on how to solve using power series, and who better than you!

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can you please show that Lim x->0 ln(x) = -infinity by the epsilon delta definition thank! :)
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I am currently a bit stuck trying to solve this problem only using the Lambert W function and elementary functions. I succeeded in solving simpler functions like y = x^x and y = x^(x^2), but I cannot seem to solve this one. Here is my current best attempt (based off of the steps for x^(x^2)): y = x^(x^x) ln(y) = x^x * ln(x) = x^x * ln(x^(x / x)) = x^x * (1/x * ln(x^x)) = 1/x * x^x * ln(x^x) x * ln(y) = x^x * ln(x^x) = ln(x^x) * x^x = ln(x^x) * e^(ln(x^x)) = x*ln(x) * e^(x*ln(x)) x*ln(x) = W(x * ln(y)) x*e^x = e^W(x * ln(y)) x = W(e^W(x * ln(y))) But this isn't particularly useful as x is still in the formula. I could solve it if I knew how to solve ln(y) = 1/x * x^x * ln(x^x). Can you please help me?
• ## A little power series.

can anyone help me find the sum of this power series?