54.4.51 problem 1509

Internal problem ID [12773]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1509
Date solved : Friday, October 03, 2025 at 03:47:23 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime \prime }+\left (4 x^{3}+\left (-4 \nu ^{2}+1\right ) x \right ) y^{\prime }+\left (4 \nu ^{2}-1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

ode:=x^3*diff(diff(diff(y(x),x),x),x)+(4*x^3+(-4*nu^2+1)*x)*diff(y(x),x)+(4*nu^2-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = x \left (c_1 \operatorname {BesselJ}\left (\nu , x\right )^{2}+c_2 \operatorname {BesselY}\left (\nu , x\right )^{2}+c_3 \operatorname {BesselJ}\left (\nu , x\right ) \operatorname {BesselY}\left (\nu , x\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to x \left (c_1 \operatorname {BesselJ}(\nu ,x)^2+c_3 \operatorname {BesselY}(\nu ,x)^2+c_2 \operatorname {BesselJ}(\nu ,x) \operatorname {BesselY}(\nu ,x)\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-4*nu**2*y(x) - x**3*Derivative(y(x), (x, 3)) + y(x))/(x*(-4*nu**2 + 4*x**2 + 1)) cannot be solved by the factorable group method