Internal problem ID [9383]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1048.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-4 y^{\prime } x +\left (3 x^{2}+2 n -1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 43
dsolve(diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(3*x^2+2*n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{\frac {x^{2}}{2}} \operatorname {KummerM}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right )+c_{2} x \,{\mathrm e}^{\frac {x^{2}}{2}} \operatorname {KummerU}\left (-\frac {n}{2}+\frac {1}{2}, \frac {3}{2}, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 37
DSolve[(-1 + 2*n + 3*x^2)*y[x] - 4*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {x^2}{2}} \left (c_1 \operatorname {HermiteH}(n,x)+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {n}{2},\frac {1}{2},x^2\right )\right ) \]