Solve the coordinate equation x* (d^2y/dx^2) + dy/ dx = x^2

You need to consider the derivative `(dy)/(dx)` as a
function `g'(x)` and x as the function `f(x)` , such
that:

`f(x)*(g'(x))' + f'(x)*g(x) =
x^2`

You should notice that the left side summation
represents the product rule, that is used when you need to evaluate the derivative of
the product of two functions, such that:

`f(x)*(g'(x))' +
f'(x)*g(x) = (f(x)*g'(x))'`

Hence, the equation you need to
solve is the following, such that:

`(f(x)*g'(x))' =
x^2`

You need to integrate both sides with respect to x,
such that:

`int (f(x)*g'(x))' dx = int x^2
dx`

`f(x)*g'(x) = x^3/3 +
c`

Replacing back x for f(x)
yields:

`x*g'(x) = x^3/3 +
c`

Dividing both sides by x
yields:

`g'(x) = (x^3/3 + c)/x => g'(x) = x^2/3 +
c/x`

You need to integrate both sides with respect ro x,
such that:

`int g'(x) dx = int x^2/3 dx + int c/x
dx`

`g(x) = y(x) = x^3/ 9 + c_1*ln x +
c_2`

Hence, evaluating the general solution
to the given second order linear ordinary differential equation, yields
`y(x) = x^3/ 9 + c_1*ln |x| + c_2.`