Internal problem ID [9088]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1509.
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}+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 30
dsolve(x^3*diff(diff(diff(y(x),x),x),x)+(4*x^3+(-4*nu^2+1)*x)*diff(y(x),x)+(4*nu^2-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \BesselJ \left (\nu , x\right )^{2}+c_{2} x \BesselY \left (\nu , x\right )^{2}+c_{3} x \BesselJ \left (\nu , x\right ) \BesselY \left (\nu , x\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 33
DSolve[(-1 + 4*nu^2)*y[x] + ((1 - 4*nu^2)*x + 4*x^3)*y'[x] + x^3*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x \left (c_1 J_{\nu }(x){}^2+c_3 Y_{\nu }(x){}^2+c_2 J_{\nu }(x) Y_{\nu }(x)\right ) \\ \end{align*}