Internal problem ID [9998]
Internal file name [OUTPUT/8945_Monday_June_06_2022_06_01_33_AM_19396141/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1677 (book 6.86).
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.
\[ \boxed {x^{2} y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}=b \,x^{2}} \]
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 trying symmetries linear in x and y(x) `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym Try integration with the canonical coordinates of the symmetry [0, x] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(_a^2*_b(_a)^2*a+2*_b(_a)-b)/_a, _b(_a), explicit` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying Riccati trying Riccati sub-methods: trying Riccati to 2nd Order -> Calling odsolve with the ODE`, diff(diff(y(x), x), x) = -(diff(y(x), x))/x+y(x)*a*b, y(x)` *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type 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 <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful <- Riccati to 2nd Order successful <- differential order: 2; canonical coordinates successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 75
dsolve(x^2*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2-b*x^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-\sqrt {-a b}\, \left (\int \frac {\operatorname {BesselY}\left (1, \sqrt {-a b}\, x \right ) c_{1} +\operatorname {BesselJ}\left (1, \sqrt {-a b}\, x \right )}{x \left (c_{1} \operatorname {BesselY}\left (0, \sqrt {-a b}\, x \right )+\operatorname {BesselJ}\left (0, \sqrt {-a b}\, x \right )\right )}d x \right )+c_{2} a \right ) x}{a} \]
✓ Solution by Mathematica
Time used: 120.409 (sec). Leaf size: 118
DSolve[-(b*x^2) + a*(-y[x] + x*y'[x])^2 + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (\int _1^x\frac {i \sqrt {b} \left (\operatorname {BesselY}\left (1,-i \sqrt {a} \sqrt {b} K[1]\right )-\operatorname {BesselJ}\left (1,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right )}{\sqrt {a} \left (\operatorname {BesselY}\left (0,-i \sqrt {a} \sqrt {b} K[1]\right )+\operatorname {BesselJ}\left (0,i \sqrt {a} \sqrt {b} K[1]\right ) c_1\right ) K[1]}dK[1]+c_2\right ) \]