60.4.57 problem 1515

Internal problem ID [11480]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1515
Date solved : Sunday, March 30, 2025 at 08:23:04 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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