Internal problem ID [9897]
Internal file name [OUTPUT/8844_Monday_June_06_2022_05_37_04_AM_45927544/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1575.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_high_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime \prime } \sin \left (x \right )^{6}+4 y^{\prime \prime \prime } \sin \left (x \right )^{5} \cos \left (x \right )-6 y^{\prime \prime } \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}=f} \] Unable to solve this ODE.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}\frac {d}{d x}y^{\prime \prime }\right ) \sin \left (x \right )^{6}+4 \left (\frac {d}{d x}y^{\prime \prime }\right ) \sin \left (x \right )^{5} \cos \left (x \right )-6 \left (\frac {d}{d x}y^{\prime }\right ) \sin \left (x \right )^{6}-4 y^{\prime } \sin \left (x \right )^{5} \cos \left (x \right )+y \sin \left (x \right )^{6}=f \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & \frac {d}{d x}\frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable Equation is the 3rd symmetric power of diff(diff(y(x),x),x)+2/3*cos(x)/sin(x)*diff(y(x),x)-1/9*(2*cos(x)^2+3*sin(x)^2)/sin(x)^2*y(x) -> Attempting now to solve this lower order ODE trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 699
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: 7.987 (sec). Leaf size: 123
DSolve[-f[x] + Sin[x]^6*y[x] - 4*Cos[x]*Sin[x]^5*y'[x] - 6*Sin[x]^6*y''[x] + 4*Cos[x]*Sin[x]^5*Derivative[3][y][x] + Sin[x]^6*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \csc (x) \left (x^3 \int _1^x\frac {1}{6} \csc ^5(K[4]) f(K[4])dK[4]+x^2 \int _1^x-\frac {1}{2} \csc ^5(K[3]) f(K[3]) K[3]dK[3]+x \int _1^x\frac {1}{2} \csc ^5(K[2]) f(K[2]) K[2]^2dK[2]+\int _1^x-\frac {1}{6} \csc ^5(K[1]) f(K[1]) K[1]^3dK[1]+c_4 x^3+c_3 x^2+c_2 x+c_1\right ) \]