Internal problem ID [9992]
Internal file name [OUTPUT/8939_Monday_June_06_2022_06_00_31_AM_61958197/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1671 (book 6.80).
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.
\[ \boxed {x y^{\prime \prime }+a \left (x y^{\prime }-y\right )^{2}=b} \]
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*_b(_a)^2*_a^4+2*_b(_a)*_a-b)/_a^2, _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 <- Chini successful <- differential order: 2; canonical coordinates successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 40
dsolve(x*diff(diff(y(x),x),x)+a*(x*diff(y(x),x)-y(x))^2-b=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (i \sqrt {b}\, \left (\int \frac {\tan \left (-i \sqrt {a}\, \sqrt {b}\, x +c_{1} \right )}{x^{2}}d x \right )+c_{2} \sqrt {a}\right ) x}{\sqrt {a}} \]
✓ Solution by Mathematica
Time used: 128.353 (sec). Leaf size: 50
DSolve[-b + a*(-y[x] + x*y'[x])^2 + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (\int _1^x\frac {\sqrt {-\frac {b}{a}} \tan \left (c_1+\frac {b K[2]}{\sqrt {-\frac {b}{a}}}\right )}{K[2]^2}dK[2]+c_2\right ) \]