60.5.35 problem 1572

Internal problem ID [11532]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1572
Date solved : Thursday, March 13, 2025 at 08:53:40 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right )^{2} y^{\prime \prime \prime \prime }+10 x \left (x^{2}-1\right ) y^{\prime \prime \prime }+\left (24 x^{2}-8-2 \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )\right ) \left (x^{2}-1\right )\right ) y^{\prime \prime }-6 x \left (\mu \left (\mu +1\right )+\nu \left (\nu +1\right )-2\right ) y^{\prime }+\left (\left (\mu \left (\mu +1\right )-\nu \left (\nu +1\right )\right )^{2}-2 \mu \left (\mu +1\right )-2 \nu \left (\nu +1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.300 (sec). Leaf size: 35

ode:=(x^2-1)^2*diff(diff(diff(diff(y(x),x),x),x),x)+10*x*(x^2-1)*diff(diff(diff(y(x),x),x),x)+(24*x^2-8-2*(mu*(mu+1)+nu*(nu+1))*(x^2-1))*diff(diff(y(x),x),x)-6*x*(mu*(mu+1)+nu*(nu+1)-2)*diff(y(x),x)+((mu*(mu+1)-nu*(nu+1))^2-2*mu*(mu+1)-2*nu*(nu+1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_{2} +c_{1} \operatorname {LegendreP}\left (\mu , x\right )\right ) \operatorname {LegendreP}\left (\nu , x\right )+\operatorname {LegendreQ}\left (\nu , x\right ) \left (\operatorname {LegendreQ}\left (\mu , x\right ) c_4 +\operatorname {LegendreP}\left (\mu , x\right ) c_3 \right ) \]

✗ Mathematica

ode=(-2*\[Mu]*(1 + \[Mu]) - 2*\[Nu]*(1 + \[Nu]) + (\[Mu]*(1 + \[Mu]) - \[Nu]*(1 + \[Nu]))^2)*y[x] - 6*(-2 + \[Mu]*(1 +\[Mu]) + \[Nu]*(1 + \[Nu]))*x*D[y[x],x] + (-8 + 24*x^3 - 2*(\[Mu]*(1 + \[Mu]) + \[Nu]*(1 + \[Nu]))*(-1 + x^2))*D[y[x],{x,2}] + 10*x*(-1 + x^2)*Derivative[3][y][x] + (-1 + x^2)^2*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
mu = symbols("mu") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(10*x*(x**2 - 1)*Derivative(y(x), (x, 3)) - 6*x*(mu*(mu + 1) + nu*(nu + 1) - 2)*Derivative(y(x), x) + (x**2 - 1)**2*Derivative(y(x), (x, 4)) + (24*x**2 - (x**2 - 1)*(2*mu*(mu + 1) + 2*nu*(nu + 1)) - 8)*Derivative(y(x), (x, 2)) + (-2*mu*(mu + 1) - 2*nu*(nu + 1) + (mu*(mu + 1) - nu*(nu + 1))**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (mu**4*y(x) + 2*mu**3*y(x) - 2*mu**2*nu**2*y(x) - 2*mu**2*nu*y(x) - 2*mu**2*x**2*Derivative(y(x), (x, 2)) - mu**2*y(x) + 2*mu**2*Derivative(y(x), (x, 2)) - 2*mu*nu**2*y(x) - 2*mu*nu*y(x) - 2*mu*x**2*Derivative(y(x), (x, 2)) - 2*mu*y(x) + 2*mu*Derivative(y(x), (x, 2)) + nu**4*y(x) + 2*nu**3*y(x) - 2*nu**2*x**2*Derivative(y(x), (x, 2)) - nu**2*y(x) + 2*nu**2*Derivative(y(x), (x, 2)) - 2*nu*x**2*Derivative(y(x), (x, 2)) - 2*nu*y(x) + 2*nu*Derivative(y(x), (x, 2)) + x**4*Derivative(y(x), (x, 4)) + 10*x**3*Derivative(y(x), (x, 3)) + 24*x**2*Derivative(y(x), (x, 2)) - 2*x**2*Derivative(y(x), (x, 4)) - 10*x*Derivative(y(x), (x, 3)) - 8*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)))/(6*x*(mu**2 + mu + nu**2 + nu - 2)) cannot be solved by the factorable group method