Internal problem ID [9048]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1469.
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 }+3 a x y^{\prime \prime }+3 a^{2} x^{2} y^{\prime }+a^{3} x^{3} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 37
dsolve(diff(diff(diff(y(x),x),x),x)+3*a*x*diff(diff(y(x),x),x)+3*a^2*x^2*diff(y(x),x)+a^3*x^3*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{-\frac {a \,x^{2}}{2}} \left (c_{1}+c_{2} {\mathrm e}^{\sqrt {3}\, \sqrt {a}\, x}+c_{3} {\mathrm e}^{-\sqrt {3}\, \sqrt {a}\, x}\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 68
DSolve[a^3*x^3*y[x] + 3*a^2*x^2*y'[x] + 3*a*x*y''[x] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {a x^2}{2}-\sqrt {3} \sqrt {a} x} \left (c_1 e^{\sqrt {3} \sqrt {a} x}+c_3 e^{2 \sqrt {3} \sqrt {a} x}+c_2\right ) \\ \end{align*}