54.4.42 problem 1498

Internal problem ID [12764]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1498
Date solved : Friday, October 03, 2025 at 03:47:20 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 53

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

\[ y = c_1 \,x^{n +\frac {1}{2}} \operatorname {BesselJ}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_2 \,x^{n +\frac {1}{2}} \operatorname {BesselY}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_3 \left (a \,x^{2}+4 n -2\right ) \]

✓ Mathematica. Time used: 3.538 (sec). Leaf size: 353

ode=-2*a*x*y[x] + (6*n + a*x^2)*D[y[x],x] - 2*(1 + n)*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 2^{-n-\frac {3}{2}} \left (\pi c_3 4^n x^4 \sec (\pi n) \operatorname {Gamma}\left (\frac {3}{2}-n\right ) \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} \operatorname {BesselJ}\left (n+\frac {1}{2},\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 {\operatorname {BesselY}\left (n+\frac {1}{2},\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}} \operatorname {Gamma}\left (n+\frac {3}{2}\right ) \left (2 a c_2 x^{n+\frac {1}{2}}-\pi \sqrt {a} c_3 x^3 \operatorname {BesselJ}\left (n-\frac {1}{2},\sqrt {a} x\right )-2 \pi c_3 x^2 \operatorname {BesselJ}\left (n-\frac {3}{2},\sqrt {a} x\right )\right )\right )+\operatorname {BesselJ}\left (n+\frac {1}{2},\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}} \operatorname {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) \operatorname {BesselJ}\left (n-\frac {1}{2},\sqrt {a} x\right )-2 \pi c_3 x^2 \tan (\pi n) \operatorname {BesselJ}\left (n-\frac {3}{2},\sqrt {a} x\right )\right )\right )}{a^2 \operatorname {Gamma}\left (n+\frac {3}{2}\right )}\right ) \end{align*}

✗ Sympy

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

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