Internal problem ID [10151]
Internal file name [OUTPUT/9098_Monday_June_06_2022_06_35_09_AM_2174505/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1830 (book 6.239).
ODE order: 2.
ODE degree: 2.
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.
\[ \boxed {3 {y^{\prime \prime }}^{2} x^{2}-2 \left (3 x y^{\prime }+y\right ) y^{\prime \prime }+4 {y^{\prime }}^{2}=0} \]
Maple trace
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation <- 2nd order ODE linearizable_by_differentiation successful ------------------- * Tackling next ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation <- 2nd order ODE linearizable_by_differentiation successful -> Calling odsolve with the ODE`, (diff(y(x), x))^2 = -(1/3)*(-6*y(x)*(diff(y(x), x))*x-y(x)^2)/x^2, y(x), singsol = none` *** Sub Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying homogeneous B trying homogeneous types: trying homogeneous D <- homogeneous successful`
✓ Solution by Maple
Time used: 0.266 (sec). Leaf size: 37
dsolve(3*x^2*diff(diff(y(x),x),x)^2-2*(3*x*diff(y(x),x)+y(x))*diff(diff(y(x),x),x)+4*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= x^{1+\frac {2 \sqrt {3}}{3}} c_{1} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {c_{1}^{2} x^{2}}{c_{2}}+c_{1} x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.028 (sec). Leaf size: 29
DSolve[4*y'[x]^2 - 2*(y[x] + 3*x*y'[x])*y''[x] + 3*x^2*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1{}^2 x^2}{c_2}+c_1 x+c_2 \\ y(x)\to \text {Indeterminate} \\ \end{align*}