Solve this cauchy differential equation:
x^2y'' + xy' + y = (lnx)^2+2x

For the corresponding homogeneous equation,

we can look for a solution of the form , with derivatives and . Substituting these into the ODE gives

which admits two solutions, and , which we can write as

and by the same token,

so we see two independent solutions that make up the characteristic solution,

For the non-homogeneous ODE, we make the substitution

so that by the chain rule, the first derivative becomes

Let . Then the second derivative becomes

Substituting these into the ODE gives

Look for a particular solution , which has second derivative . Substituting these into the ODE gives

so that the particular solution is

Solving in terms of gives the solution

and the overall general solution is

it 2

step-by-step explanation:

8^1/3=2

ion

step-by-step explanation:

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