54.4.72 problem 1530

Internal problem ID [12794]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1530
Date solved : Friday, October 03, 2025 at 03:47:26 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 105

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_2 \cos \left (x \right )^{2} \operatorname {hypergeom}\left (\left [1+\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_3 \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \cos \left (x \right ) \operatorname {hypergeom}\left (\left [1+\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to c_3 \operatorname {LegendreP}(\nu ,\cos (x)) \operatorname {LegendreQ}(\nu ,\cos (x))+c_1 \operatorname {LegendreP}(\nu ,\cos (x))^2+c_2 \operatorname {LegendreQ}(\nu ,\cos (x))^2 \end{align*}

✗ Sympy

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

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