Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$.

Question

Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$.

Answer 1

Well, for starters $$ y(x)= \int_x^\infty e^{-t^2}dt=-\int^x_\infty e^{-t^2}dt $$ so by the fundamental theorem of calculus $$ y'(x)=-e^{-x^2}, y''(x)=2x e^{-x^2} $$

and it is easy to check that $$ y''+2xy=0 $$