Internal problem ID [9760]
Internal file name [OUTPUT/8702_Monday_June_06_2022_05_15_21_AM_63797037/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1433.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\frac {\sin \left (x \right ) y^{\prime }}{\cos \left (x \right )}+\frac {\left (2 x^{2}+x^{2} \sin \left (x \right )^{2}-24 \cos \left (x \right )^{2}\right ) y}{4 x^{2} \cos \left (x \right )^{2}}=\sqrt {\cos \left (x \right )}} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] trying symmetries linear in x and y(x) -> Try solving first the homogeneous part of the 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 <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve(diff(diff(y(x),x),x) = -sin(x)/cos(x)*diff(y(x),x)-1/4*(2*x^2+x^2*sin(x)^2-24*cos(x)^2)/x^2/cos(x)^2*y(x)+cos(x)^(1/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sqrt {\cos \left (x \right )}\, \left (4 c_{1} x^{5}-x^{4}+4 c_{2} \right )}{4 x^{2}} \]
✓ Solution by Mathematica
Time used: 0.22 (sec). Leaf size: 35
DSolve[y''[x] == Sqrt[Cos[x]] - (Sec[x]^2*(2*x^2 - 24*Cos[x]^2 + x^2*Sin[x]^2)*y[x])/(4*x^2) - Tan[x]*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (4 c_2 x^5-5 x^4+20 c_1\right ) \sqrt {\cos (x)}}{20 x^2} \]