I was just doing a question from my textbook. I am practising for my test. The question is to solve a differential equation
$\displaystyle\frac{dy}{dx}+ \frac{y}{x} + 1 = 5x$, $y(0) = 1.$
The answer that i have come up with is
$(xy+y)= 5x^3/3+5x^2/2+c$
by substituting the values $x=0$ and $y=1$ in to the general equation I get
$y(x+1)=5x^3/3 + 5x^2/2 +1$
as the particular solution.
Can you tell me how will the particular solution look like and why this particular solution exists?