Internal problem ID [9045]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1468.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime } x +2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 y a x=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 64
dsolve(diff(diff(diff(y(x),x),x),x)-6*x*diff(diff(y(x),x),x)+2*(4*x^2+2*a-1)*diff(y(x),x)-8*y(x)*a*x=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}+c_{2} x^{2} \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2}+c_{3} x^{2} \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.041 (sec). Leaf size: 57
DSolve[-8*a*x*y[x] + 2*(-1 + 2*a + 4*x^2)*y'[x] - 6*x*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \operatorname {HermiteH}\left (\frac {a}{2},x\right ) \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )+c_1 \operatorname {HermiteH}\left (\frac {a}{2},x\right )^2+c_3 \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )^2 \\ \end{align*}