Internal problem ID [9839]
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]]
\[ \boxed {x^{3} y^{\prime \prime \prime }+6 x^{2} y^{\prime \prime }+\left (a \,x^{3}-12\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 134
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)
\[ y \left (x \right ) = \frac {-c_{2} \left (\left (-i-\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) {\mathrm e}^{-\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) x}{2 a}}+c_{3} \left (\left (-i+\sqrt {3}\right ) \left (-a^{4}\right )^{\frac {2}{3}}+i x \,a^{3}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) x}{2 a}}+c_{1} \left (x \,a^{3}+2 \left (-a^{4}\right )^{\frac {2}{3}}\right ) {\mathrm e}^{\frac {\left (-a^{4}\right )^{\frac {1}{3}} x}{a}}}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.366 (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]
\[ y(x)\to \frac {c_1 e^{-\sqrt [3]{a} x} \left (\sqrt [3]{a} x+2\right )+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x} \left (\sqrt [3]{a} x+2 (-1)^{2/3}\right )+c_3 e^{-(-1)^{2/3} \sqrt [3]{a} x} \left (\sqrt [3]{a} x-2 \sqrt [3]{-1}\right )}{x^3} \]