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

the equivalent expression are always the hardest ones but you look at the gcf also

step-by-step explanation:

its the second choice.

step-by-step explanation:

the elements in the range will be 7 less than the elements in the domain.