Internal problem ID [4674]
Internal file name [OUTPUT/4167_Sunday_June_05_2022_12_32_17_PM_91608857/index.tex]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 8. Special second order equations. Lesson 35. Independent variable x
absent
Problem number: Exercise 35.23(c), page 504.
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], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {x y y^{\prime \prime }-2 x {y^{\prime }}^{2}+\left (y+1\right ) y^{\prime }=0} \]
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 trying symmetries linear in x and y(x) `, `-> Computing symmetries using: way = 3 Try integration with the canonical coordinates of the symmetry [x, 0] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = _b(_a)*(_b(_a)-2)/_a, _b(_a), explicit, HINT = [[_a, 0]]` *** Sublevel 2 *** symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[_a, 0]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 22
dsolve(x*y(x)*diff(y(x),x$2)-2*x*(diff(y(x),x))^2+(1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 20.549 (sec). Leaf size: 52
DSolve[x*y[x]*y''[x]-2*x*(y'[x])^2+(1+y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}} \\ y(x)\to \frac {1}{2} (\log (x)-c_2) \\ \end{align*}