Internal problem ID [9969]
Internal file name [OUTPUT/8916_Monday_June_06_2022_05_50_54_AM_50578732/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1647 (6.57).
ODE order: 2.
ODE degree: 0.
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.
\[ \boxed {y^{\prime \prime }-a \left (x y^{\prime }-y\right )^{v}=0} \]
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 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 Try integration with the canonical coordinates of the symmetry [0, x] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -(-a*(_a^2*_b(_a))^v+2*_b(_a))/_a, _b(_a), explicit, HINT = [[_a*(v-1)/(v+1), - symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a*(v-1)/(v+1), -2*_b*v/(v+1)]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 59
dsolve(diff(diff(y(x),x),x)-a*(x*diff(y(x),x)-y(x))^v=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (2^{\frac {1}{v -1}} \left (\int -\frac {\left (\left (v -1\right ) a \,x^{2}-c_{1} \right ) \left (-\frac {1}{\left (v -1\right ) a \,x^{2}-c_{1}}\right )^{\frac {v}{v -1}}}{x^{2}}d x \right )+c_{2} \right ) x \]
✓ Solution by Mathematica
Time used: 120.631 (sec). Leaf size: 60
DSolve[-(a*(-y[x] + x*y'[x])^v) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (\int _1^x\left (\frac {1}{2} a K[2]^{2 v}-\frac {1}{2} a v K[2]^{2 v}+c_1 K[2]^{2 v-2}\right ){}^{\frac {1}{1-v}}dK[2]+c_2\right ) \]