23.4.232 problem 232

Internal problem ID [6534]
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 : 232
Date solved : Tuesday, September 30, 2025 at 03:02:47 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \sqrt {a^{2}+x^{2}}\, \left (b {y^{\prime }}^{2}+y y^{\prime \prime }\right )&=y y^{\prime } \end{align*}

✓ Maple. Time used: 0.071 (sec). Leaf size: 72

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

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

✓ Mathematica. Time used: 61.89 (sec). Leaf size: 177

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

\begin{align*} y(x)&\to c_2 \exp \left (\int _1^x\frac {2 a^2 e^{\text {arctanh}\left (\frac {K[1]}{\sqrt {a^2+K[1]^2}}\right )}}{(b+1) \log \left (a \left (\sqrt {\frac {a^2}{a^2+K[1]^2}} a^2+K[1] a+K[1]^2 \sqrt {\frac {a^2}{a^2+K[1]^2}}\right )\right ) a^3+\left (2 c_1+(b+1) K[1] \sqrt {\frac {a^2}{a^2+K[1]^2}}\right ) a^2+(b+1) K[1]^2 \sqrt {\frac {a^2}{a^2+K[1]^2}} \left (K[1]+\sqrt {a^2+K[1]^2}\right )}dK[1]\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt((-4*a**2*b*Derivative(y(x), (x, 2))