54.5.37 problem 1574

Internal problem ID [12833]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1574
Date solved : Friday, October 03, 2025 at 03:47:32 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.337 (sec). Leaf size: 204

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

\[ y = \sin \left (x \right ) \left (c_1 \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+c_2 \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {3}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos \left (x \right )^{2}\right )+\cos \left (x \right ) \left (\operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right ) c_3 +\operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-a^{4}+1}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos \left (x \right )^{2}\right ) c_4 \right )\right ) \]

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 265

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

\begin{align*} y(x)&\to \sin (x) \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (3-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+3\right ),\frac {1}{2},\cos ^2(x)\right )+c_3 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (5-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+5\right ),\frac {3}{2},\cos ^2(x)\right )+c_2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (3-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+3\right ),\frac {1}{2},\cos ^2(x)\right )+c_4 \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (5-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+5\right ),\frac {3}{2},\cos ^2(x)\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a**4*(1 - cos(2*x))**2*y(x) - (1 - cos(2*x))**2*Derivative(y(x), (x, 2)) - (1 - cos(2*x))**2*Derivative(y(x), (x, 4)) + 12*y(x) - 2*sin(2*x)*Derivative(y(x), (x, 3)) + sin(4*x)*Derivative(y(x), (x, 3)) - 6*cos(2*x)*Derivative(y(x), (x, 2)) + 6*Derivative(y(x), (x, 2)))/(8*sin(2*x) - sin(4*x)) cannot be solved by the factorable group method