Internal problem ID [8629]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1049.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}-1\right ) y-{\mathrm e}^{x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 70
dsolve(diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(4*x^2-1)*y(x)-exp(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{x^{2}} \cos \relax (x ) c_{2}+{\mathrm e}^{x^{2}} \sin \relax (x ) c_{1}-\frac {{\mathrm e}^{x^{2}} \left (-{\mathrm e}^{\frac {i}{2}} \left (i \cos \relax (x )+\sin \relax (x )\right ) \erf \left (x -\frac {1}{2}-\frac {i}{2}\right )+\erf \left (x -\frac {1}{2}+\frac {i}{2}\right ) \left (i \cos \relax (x )-\sin \relax (x )\right ) {\mathrm e}^{-\frac {i}{2}}\right ) \sqrt {\pi }}{4} \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 102
DSolve[-E^x + (-1 + 4*x^2)*y[x] - 4*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{x (x-i)-\frac {i}{2}} \left (\sqrt {\pi } \left (e^{2 i x} \text {Erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right )-i x\right )-i e^i \text {Erf}\left (-x+\left (\frac {1}{2}+\frac {i}{2}\right )\right )\right )+2 e^{\frac {i}{2}} \left (2 c_1-i c_2 e^{2 i x}\right )\right ) \\ \end{align*}