Solve the initial value problem 2ty" + 10ty' + 8y = 0, for t > 0, y(1) = 1, y'(1) = 0.

I think you meant to write

which is an ODE of Cauchy-Euler type. Let . Then

Substituting and its derivatives into the ODE gives

Divide through by , which we can do because :

Since this root has multiplicity 2, we get the characteristic solution

If you're not sure where the logarithm comes from, scroll to the bottom for a bit more in-depth explanation.

With the given initial values, we find

so that the particular solution is

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Under the hood, we're actually substituting , so that . When we do this, we need to account for the derivative of wrt the new variable . By the chain rule,

Since is a function of , we can treat in the same way, so denote this by . By the quotient rule,

and by the chain rule,

where

so that

Plug all this into the original ODE to get a new one that is linear in with constant coefficients:

which has characteristic equation

and admits the characteristic solution

Finally replace to get the solution we found earlier,

answer: what statements

step-by-step explanation if you write the statements down then i could you.