Internal problem ID [10089]
Internal file name [OUTPUT/9036_Monday_June_06_2022_06_15_44_AM_70530116/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1768 (book 6.177).
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, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x y y^{\prime \prime }+\left (\frac {a x}{\sqrt {b^{2}-x^{2}}}-x \right ) {y^{\prime }}^{2}-y y^{\prime }=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful integrating factor(s) found: x*y(x)/diff(y(x),x)^2 attempting the computation of a related first integral... -> Calling odsolve with the ODE`, _a*_b(_a)/(diff(_b(_a), _a))+(a*b^2-a*_a^2+c__1*(-_a^2+b^2)^(1/2))/(-_a^2+b^2)^(1/2) = 0, _b(_a Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- differential order: 2; exact nonlinear successful <- 2nd_order mu_xyp successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 55
dsolve(x*y(x)*diff(diff(y(x),x),x)+(a*x/(b^2-x^2)^(1/2)-x)*diff(y(x),x)^2-y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{2} {\mathrm e}^{-\left (\int \frac {x \sqrt {b^{2}-x^{2}}}{c_{1} \sqrt {b^{2}-x^{2}}+a \left (b^{2}-x^{2}\right )}d x \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 19.437 (sec). Leaf size: 54
DSolve[-(y[x]*y'[x]) + (-x + (a*x)/Sqrt[b^2 - x^2])*y'[x]^2 + x*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{\frac {\sqrt {b^2-x^2}}{a}} \left (a \sqrt {b^2-x^2}-c_1\right ){}^{\frac {c_1}{a^2}} \]