Internal problem ID [9077]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1498.
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 }-2 \left (n +1\right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 y a x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 53
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-2*(n+1)*x*diff(diff(y(x),x),x)+(a*x^2+6*n)*diff(y(x),x)-2*y(x)*a*x=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{n +\frac {1}{2}} \BesselJ \left (-n -\frac {1}{2}, x \sqrt {a}\right )+c_{2} x^{n +\frac {1}{2}} \BesselY \left (-n -\frac {1}{2}, x \sqrt {a}\right )+c_{3} \left (a \,x^{2}+4 n -2\right ) \]
✓ Solution by Mathematica
Time used: 6.266 (sec). Leaf size: 353
DSolve[-2*a*x*y[x] + (6*n + a*x^2)*y'[x] - 2*(1 + n)*x*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2^{-n-\frac {3}{2}} \left (\pi c_3 4^n x^4 \sec (\pi n) \Gamma \left (\frac {3}{2}-n\right ) \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} J_{n+\frac {1}{2}}\left (\sqrt {a} x\right ) \, _1\tilde {F}_2\left (\frac {3}{2}-n;\frac {1}{2}-n,\frac {5}{2}-n;-\frac {a x^2}{4}\right )+\frac {Y_{n+\frac {1}{2}}\left (\sqrt {a} x\right ) \left (2 \pi c_3 \left (4 n^2-1\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}}+a 2^{n+\frac {1}{2}} \Gamma \left (n+\frac {3}{2}\right ) \left (2 a c_2 x^{n+\frac {1}{2}}-\pi \sqrt {a} c_3 x^3 J_{n-\frac {1}{2}}\left (\sqrt {a} x\right )-2 \pi c_3 x^2 J_{n-\frac {3}{2}}\left (\sqrt {a} x\right )\right )\right )+J_{n+\frac {1}{2}}\left (\sqrt {a} x\right ) \left (2 \pi c_3 \left (4 n^2-1\right ) \tan (\pi n) \left (\sqrt {a} x\right )^{n+\frac {1}{2}}+a 2^{n+\frac {1}{2}} \Gamma \left (n+\frac {3}{2}\right ) \left (2 a c_1 x^{n+\frac {1}{2}}-\pi \sqrt {a} c_3 x^3 \tan (\pi n) J_{n-\frac {1}{2}}\left (\sqrt {a} x\right )-2 \pi c_3 x^2 \tan (\pi n) J_{n-\frac {3}{2}}\left (\sqrt {a} x\right )\right )\right )}{a^2 \Gamma \left (n+\frac {3}{2}\right )}\right ) \\ \end{align*}