Internal problem ID [10150]
Internal file name [OUTPUT/9097_Monday_June_06_2022_06_34_57_AM_41125478/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1829 (book 6.238).
ODE order: 2.
ODE degree: 2.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y=0} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(-2\) is missing which is not the case here.
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 = (1/16)*x^4-(1/2)*(diff(y(x), x))*x^3+y(x)*x^2-(diff(y(x), x))*x+y(x), y(x), si 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 dAlembert trying simple symmetries for implicit equations Successful isolation of dy/dx: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(y(x), x), x)-x*(diff(y(x), x))/(x^2+1)-(1/4)*x^2/(x^2+1), y(x)` *** Sublev Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable -> Calling odsolve with the ODE`, diff(_b(_a), _a) = (1/4)*_a*(4*_b(_a)+_a)/(_a^2+1), _b(_a)` *** Sublevel 5 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful <- high order exact linear fully integrable successful <- 1st order ODE linearizable_by_differentiation successful ------------------- * Tackling next ODE. *** Sublevel 3 *** Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation <- 1st order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 59
dsolve(2*(x^2+1)*diff(diff(y(x),x),x)^2-x*diff(diff(y(x),x),x)*(x+4*diff(y(x),x))+2*(x+diff(y(x),x))*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (c_{1} +\frac {\operatorname {arcsinh}\left (x \right )}{4}\right ) x \sqrt {x^{2}+1}}{2}-\frac {3 x^{2}}{16}+c_{1}^{2}+\frac {c_{1} \operatorname {arcsinh}\left (x \right )}{2}+\frac {\operatorname {arcsinh}\left (x \right )^{2}}{16} \\ y \left (x \right ) &= \frac {1}{2} c_{1} x^{2}+c_{2} x +c_{1}^{2}+c_{2}^{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.263 (sec). Leaf size: 32
DSolve[-2*y[x] + 2*y'[x]*(x + y'[x]) - x*(x + 4*y'[x])*y''[x] + 2*(1 + x^2)*y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{2} \sqrt {c_2-c_1{}^2} x^2+c_1 x+c_2 \]