Internal problem ID [9067]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1488.
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 }+a \,x^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 148
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-6*diff(y(x),x)+a*x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (a^{3} x +2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x}+\frac {c_{2} {\mathrm e}^{-\frac {i \left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}} \left (\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}-i a^{3} x +i \left (-a^{4}\right )^{\frac {2}{3}}\right )}{x}+\frac {c_{3} {\mathrm e}^{\frac {i \left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}} \left (i a^{3} x +\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}-i \left (-a^{4}\right )^{\frac {2}{3}}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.278 (sec). Leaf size: 97
DSolve[a*x^2*y[x] - 6*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} \left (\sqrt [3]{a} x+2\right )+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x} \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x} \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )}{x} \\ \end{align*}