Internal problem ID [9382]
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]]
\[ \boxed {y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-1\right ) y={\mathrm e}^{x}} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 66
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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{x^{2}} \left ({\mathrm e}^{\frac {i}{2}} \sqrt {\pi }\, \left (i \cos \left (x \right )+\sin \left (x \right )\right ) \operatorname {erf}\left (x -\frac {1}{2}-\frac {i}{2}\right )-\left (i \cos \left (x \right )-\sin \left (x \right )\right ) {\mathrm e}^{-\frac {i}{2}} \sqrt {\pi }\, \operatorname {erf}\left (x -\frac {1}{2}+\frac {i}{2}\right )+4 c_{1} \sin \left (x \right )+4 c_{2} \cos \left (x \right )\right )}{4} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 105
DSolve[-E^x + (-1 + 4*x^2)*y[x] - 4*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{4} e^{x (x-i)-\frac {i}{2}} \left (-i e^i \sqrt {\pi } \text {erf}\left (-x+\left (\frac {1}{2}+\frac {i}{2}\right )\right )+\sqrt {\pi } e^{2 i x} \text {erfi}\left (\left (\frac {1}{2}+\frac {i}{2}\right )-i x\right )+2 e^{\frac {i}{2}} \left (2 c_1-i c_2 e^{2 i x}\right )\right ) \]