60.4.44 problem 1500

Internal problem ID [11467]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1500
Date solved : Sunday, March 30, 2025 at 08:22:49 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 114

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

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