54.5.38 problem 1575

Internal problem ID [12834]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1575
Date solved : Friday, October 03, 2025 at 03:47:32 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}-f&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 115

ode:=diff(diff(diff(diff(y(x),x),x),x),x)*sin(x)^6+4*diff(diff(diff(y(x),x),x),x)*sin(x)^5*cos(x)-6*diff(diff(y(x),x),x)*sin(x)^6-4*diff(y(x),x)*sin(x)^5*cos(x)+y(x)*sin(x)^6-f = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\csc \left (x \right ) \left (-3 f \,x^{3} \ln \left (1-{\mathrm e}^{i x}\right )+3 f \,x^{3} \ln \left (1+{\mathrm e}^{i x}\right )+3 f \ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) x^{3}+48 c_4 \,x^{3}+20 i \operatorname {polylog}\left (2, {\mathrm e}^{i x}\right ) f -20 i \operatorname {polylog}\left (2, -{\mathrm e}^{i x}\right ) f +18 i \operatorname {polylog}\left (4, -{\mathrm e}^{i x}\right ) f -18 i \operatorname {polylog}\left (4, {\mathrm e}^{i x}\right ) f +48 c_3 \,x^{2}+2 f \csc \left (x \right )+48 c_2 x +48 c_1 \right )}{48} \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 123

ode=-f[x] + Sin[x]^6*y[x] - 4*Cos[x]*Sin[x]^5*D[y[x],x] - 6*Sin[x]^6*D[y[x],{x,2}] + 4*Cos[x]*Sin[x]^5*Derivative[3][y][x] + Sin[x]^6*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \csc (x) \left (x^3 \int _1^x\frac {1}{6} \csc ^5(K[4]) f(K[4])dK[4]+x^2 \int _1^x-\frac {1}{2} \csc ^5(K[3]) f(K[3]) K[3]dK[3]+x \int _1^x\frac {1}{2} \csc ^5(K[2]) f(K[2]) K[2]^2dK[2]+\int _1^x-\frac {1}{6} \csc ^5(K[1]) f(K[1]) K[1]^3dK[1]+c_4 x^3+c_3 x^2+c_2 x+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
f = symbols("f") 
y = Function("y") 
ode = Eq(-f + y(x)*sin(x)**6 - 6*sin(x)**6*Derivative(y(x), (x, 2)) + sin(x)**6*Derivative(y(x), (x, 4)) - 4*sin(x)**5*cos(x)*Derivative(y(x), x) + 4*sin(x)**5*cos(x)*Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE f/(4*sin(x)**5*cos(x)) - y(x)*tan(x)/4 + 3*tan(x)*Derivative(y(x), (x, 2))/2 - tan(x)*Derivative(y(x), (x, 4))/4 + Derivative(y(x), x) - Derivative(y(x), (x, 3)) cannot be solved by the factorable group method