Internal problem ID [9071]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1492.
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 }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 39
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-3*(x-m)*x*diff(diff(y(x),x),x)+(2*x^2+4*(n-m)*x+m*(2*m-1))*diff(y(x),x)-2*n*(2*x-2*m+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \KummerM \left (-n , m , x\right )^{2}+c_{2} \KummerU \left (-n , m , x\right )^{2}+c_{3} \KummerM \left (-n , m , x\right ) \KummerU \left (-n , m , x\right ) \]
✓ Solution by Mathematica
Time used: 0.19 (sec). Leaf size: 43
DSolve[-2*n*(1 - 2*m + 2*x)*y[x] + (m*(-1 + 2*m) + 4*(-m + n)*x + 2*x^2)*y'[x] - 3*x*(-m + x)*y''[x] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 L_n^{m-1}(x) \text {HypergeometricU}(-n,m,x)+c_1 \text {HypergeometricU}(-n,m,x)^2+c_3 L_n^{m-1}(x){}^2 \\ \end{align*}