23.4.241 problem 241

Internal problem ID [6543]
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 : 241
Date solved : Tuesday, September 30, 2025 at 03:04:02 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (1+y^{2}\right ) y^{\prime \prime }&=\left (a +3 y\right ) {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 50

ode:=(1+y(x)^2)*diff(diff(y(x),x),x) = (a+3*y(x))*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i \\ y &= i \\ \frac {\left (y-a \right ) {\mathrm e}^{-a \arctan \left (y\right )}}{\sqrt {1+y^{2}}\, \left (a^{2}+1\right )}-c_1 x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.526 (sec). Leaf size: 141

ode=(1 + y[x]^2)*D[y[x],{x,2}] == (a + 3*y[x])*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) (-c_1)}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {(\text {$\#$1}-a) e^{-a \arctan (\text {$\#$1})}}{\sqrt {\text {$\#$1}^2+1} \left (a^2+1\right ) c_1}\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-(a + 3*y(x))*Derivative(y(x), x)**2 + (y(x)**2 + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((y(x)**2 + 1)*Derivative(y(x), (x, 2))/(a + 3*y(x))) + Der