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

You need to consider the derivative as a
function and x as the function , such
that:

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))'

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

x^2

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

dx

c

Replacing back x for f(x)
yields:

c

Dividing both sides by x
yields:

c/x

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

dx

c_2

Hence, evaluating the general solution
to the given second order linear ordinary differential equation, yields