60.5.40 problem 1575

Internal problem ID [11574]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1575
Date solved : Tuesday, January 28, 2025 at 06:06:56 PM
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*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 643

dsolve(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,y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 0.183 (sec). Leaf size: 123

DSolve[-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,y[x],x,IncludeSingularSolutions -> True]
 

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