Converting integral equation to its primary initial value problem

Question

I converted below initial value problem to Volterra equation of second kind $$ y'(x)-2xy(x)=e^{x^2}, \hspace{3mm} y(0)=1 $$ Supposing $u(x)=y'(x)$ and integrating both sides from $0$ to $x$ yields the Volterra equation corresponding to the above differential equation $$ u(x)=2x+e^{x^2}+2x\int_{0}^{x}u(t)\,dt $$ Now I tried to check whether my answer is right. I differentiated both sides with respect to $x$ I got $$ u'(x)=2+2x\,e^{x^2}+2\int_{0}^{x}u(t)\,dt+2x\,u(x) $$ Rewriting this in form of differential equation above we have $$ u'(x)-2xu(x)=2+2x\,e^{x^2}+2\int_{0}^{x}u(t)\,dt \,,\hspace{3mm} u(0)=1 $$ But now the above differential equation I have gotten is not equal to the primary initial value problem. What's exactly wrong?

Answer 1

You did nothing wrong. It is not expected to obtain the same equation since you are expressing the result using another function. The equation you obtained is the derivative of the original equation.