Internal problem ID [9075]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1496.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime \prime }+6 y^{\prime \prime } x +6 y^{\prime }+a \,x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
dsolve(x^2*diff(diff(diff(y(x),x),x),x)+6*x*diff(diff(y(x),x),x)+6*diff(y(x),x)+a*x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\left (-\frac {\left (-a \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (-a \right )^{\frac {1}{3}}}{2}\right ) x}+c_{2} {\mathrm e}^{\left (-\frac {\left (-a \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (-a \right )^{\frac {1}{3}}}{2}\right ) x}+c_{3} {\mathrm e}^{\left (-a \right )^{\frac {1}{3}} x}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 58
DSolve[a*x^2*y[x] + 6*y'[x] + 6*x*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 e^{-\sqrt [3]{a} x}+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x}+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x}}{x^2} \\ \end{align*}