Diff. eqn + erf (error function)

[veronik]

I’m stacked with this problem for many days, someone can help me pleeeeease:

(a) $$f \left( x \right) =\int _{-\infty }^{{x}^{2}/2}\!{e^{x-1/2\,{t}^{2 }}}{dt}$$

I foud the solution: $$f \left( x \right) =1/2\,{e^{x}}\sqrt {2\pi } \left( 1+{\it erf} \left( 1/4\,{x}^{2}\sqrt {2} \right) \right)$$

(b) Find the solution of the dfferential equatio:
$${\frac {d^{2}}{d{x}^{2}}}y \left( x \right) =f \left( x \right)$$ with y(0)=0 and dy(0)/dx = 0

In the form : $$y \left( x \right) =\int _{0}^{x}\! \left( x-t \right) f \left( t \right) {dt}$$



Veronica

 

[lippyka]

answered this correctly. The solution is:y \left( x \right) =\int _{0}^{x}\! \left( x-t \right) f \left( t \right) {dt} = 1/8\,{\sqrt {2\pi }}{e^{x}}\left( \left( 4\,x+1 \right) {\it erf} \left( 1/4\,{x}^{2}\sqrt {2} \right) -4\,{x}^{2} \right)