2.2.8 Problem 9

Internal problem ID [9131]
Book : Second order enumerated odes
Section : section 2
Problem number : 9
Date solved : Thursday, March 13, 2025 at 06:58:34 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

\begin{align*} y^{\prime \prime }+\left (3+x \right ) y^{\prime }+\left (3+y^{2}\right ) {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+(x+3)*diff(y(x),x)+(3+y(x)^2)*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ c_{1} \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +3\right )}{2}\right )-c_{2} +\int _{}^{y}{\mathrm e}^{\frac {\textit {\_a} \left (\textit {\_a}^{2}+9\right )}{3}}d \textit {\_a} = 0 \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful`
 

✓ Mathematica. Time used: 9.111 (sec). Leaf size: 56

ode=D[y[x],{x,2}]+(3+x)*D[y[x],x]+(3+y[x]^2)*(D[y[x],x])^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{\frac {K[2]^3}{3}+3 K[2]}dK[2]\&\right ]\left [\int _1^x-e^{-\frac {1}{2} K[3] (K[3]+6)} c_1dK[3]+c_2\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x + sqrt(x**2 + 6*x - 4*y(x)**2*Derivative(y(x), (x, 2)) - 12*Derivative(y(x), (x, 2)) + 9) - 3)/(2*(y(x)**2 + 3)) cannot be solved by the factorable group method