Internal problem ID [9093]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1514.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime } x^{3}+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 148
dsolve(x^3*diff(diff(diff(y(x),x),x),x)+6*x^2*diff(diff(y(x),x),x)+(a*x^3-12)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (a^{3} x +2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x^{3}}+\frac {c_{2} \left (\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}-i a^{3} x +i \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{-\frac {i \left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}}}{x^{3}}+\frac {c_{3} \left (i a^{3} x +\left (-a^{4}\right )^{\frac {2}{3}} \sqrt {3}-i \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {i \left (\sqrt {3}+i\right ) \left (-a^{4}\right )^{\frac {1}{3}} x}{2 a}}}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.382 (sec). Leaf size: 97
DSolve[(-12 + a*x^3)*y[x] + 6*x^2*y''[x] + x^3*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^3} \\ \end{align*}