54.4.36 problem 1492

Internal problem ID [12758]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1492
Date solved : Friday, October 03, 2025 at 03:47:19 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Maple. Time used: 0.106 (sec). Leaf size: 39

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {KummerM}\left (-n , m , x\right )^{2}+c_2 \operatorname {KummerU}\left (-n , m , x\right )^{2}+c_3 \operatorname {KummerM}\left (-n , m , x\right ) \operatorname {KummerU}\left (-n , m , x\right ) \]

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 43

ode=-2*n*(1 - 2*m + 2*x)*y[x] + (m*(-1 + 2*m) + 4*(-m + n)*x + 2*x^2)*D[y[x],x] - 3*x*(-m + x)*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \operatorname {HypergeometricU}(-n,m,x) L_n^{m-1}(x)+c_1 \operatorname {HypergeometricU}(-n,m,x)^2+c_3 L_n^{m-1}(x){}^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-2*n*(-2*m + 2*x + 1)*y(x) + x**2*Derivative(y(x), (x, 3)) - x*(-3*m + 3*x)*Derivative(y(x), (x, 2)) + (m*(2*m - 1) + 2*x**2 + x*(-4*m + 4*n))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-4*m*n*y(x) - 3*m*x*Derivative(y(x), (x, 2)) + 4*n*x*y(x) + 2*n*y(x) + 3*x**2*Derivative(y(x), (x, 2)) - x**2*Derivative(y(x), (x, 3)))/(2*m**2 - 4*m*x - m + 4*n*x + 2*x**2) cannot be solved by the factorable group method