60.4.52 problem 1510

Internal problem ID [11475]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1510
Date solved : Sunday, March 30, 2025 at 08:22:59 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 102

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

\[ y(x)\to c_1 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,1\right ]}{\nu }}+c_2 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,2\right ]}{\nu }}+c_3 x^{1-\nu } e^{\frac {x^{\nu } \text {Root}\left [\text {$\#$1}^3+\text {$\#$1} a+b\&,3\right ]}{\nu }} \]

✗ Sympy

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

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