54.4.49 problem 1505

Internal problem ID [12771]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1505
Date solved : Friday, October 03, 2025 at 03:47:22 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a y&=0 \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 79

ode:=2*x*(x-1)*diff(diff(diff(y(x),x),x),x)+3*(2*x-1)*diff(diff(y(x),x),x)+(2*a*x+b)*diff(y(x),x)+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {MathieuC}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_2 \operatorname {MathieuS}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_3 \operatorname {MathieuC}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \operatorname {MathieuS}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \]

✓ Mathematica. Time used: 60.132 (sec). Leaf size: 115

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

\begin{align*} y(x)&\to c_3 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ] \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]+c_1 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]^2+c_2 \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]^2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x) + 2*x*(x - 1)*Derivative(y(x), (x, 3)) + (6*x - 3)*Derivative(y(x), (x, 2)) + (2*a*x + b)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) - 2*x**2*Derivative(y(x), (x, 3)) - 6*x*Derivative(y(x), (x, 2)) + 2*x*Derivative(y(x), (x, 3)) + 3*Derivative(y(x), (x, 2)))/(2*a*x + b) cannot be solved by the factorable group method