23.4.46 problem 46
Internal
problem
ID
[6348]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
46
Date
solved
:
Tuesday, September 30, 2025 at 02:52:15 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} g \left (y\right )+f \left (y\right ) {y^{\prime }}^{2}+y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.064 (sec). Leaf size: 99
ode:=g(y(x))+f(y(x))*diff(y(x),x)^2+diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {{\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}}}{\sqrt {{\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}} \left (-2 \int {\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}} g \left (\textit {\_f} \right )d \textit {\_f} +c_1 \right )}}d \textit {\_f} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {{\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}}}{\sqrt {{\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}} \left (-2 \int {\mathrm e}^{2 \int f \left (\textit {\_f} \right )d \textit {\_f}} g \left (\textit {\_f} \right )d \textit {\_f} +c_1 \right )}}d \textit {\_f} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.252 (sec). Leaf size: 458
ode=g[y[x]] + f[y[x]]*D[y[x],x]^2 + D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[4]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}-f(K[1])dK[1]\right )}{\sqrt {2 \int _1^{K[3]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]-c_1}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}-f(K[1])dK[1]\right )}{\sqrt {2 \int _1^{K[4]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]-c_1}}dK[4]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}-f(K[1])dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\exp \left (-2 \int _1^{K[2]}-f(K[1])dK[1]\right ) g(K[2])dK[2]}}dK[4]\&\right ][x+c_2] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(f(y(x))*Derivative(y(x), x)**2 + g(y(x)) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE sqrt(-(g(y(x)) + Derivative(y(x), (x, 2)))/f(y(x))) + Derivative