29.34.4 problem 999

Internal problem ID [5575]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 999
Date solved : Sunday, March 30, 2025 at 09:03:42 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} a^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) {y^{\prime }}^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.306 (sec). Leaf size: 200

ode:=a^2*(b^2-(c*x-a*y(x))^2)*diff(y(x),x)^2+2*a*b^2*c*diff(y(x),x)+c^2*(b^2-(c*x-a*y(x))^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c x -\sqrt {2}\, b}{a} \\ y &= \frac {c x +\sqrt {2}\, b}{a} \\ y &= \frac {\operatorname {RootOf}\left (-a \int _{}^{\textit {\_Z}}\frac {a^{2} \textit {\_a}^{2}-2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (a^{2} \textit {\_a}^{2}-2 b^{2}\right )}}{a^{2} \textit {\_a}^{2}-2 b^{2}}d \textit {\_a} +2 c_1 c -2 c x \right ) a +c x}{a} \\ y &= \frac {\operatorname {RootOf}\left (a \int _{}^{\textit {\_Z}}-\frac {a^{2} \textit {\_a}^{2}-2 b^{2}-\sqrt {-a^{2} \textit {\_a}^{2} \left (a^{2} \textit {\_a}^{2}-2 b^{2}\right )}}{a^{2} \textit {\_a}^{2}-2 b^{2}}d \textit {\_a} +2 c_1 c -2 c x \right ) a +c x}{a} \\ \end{align*}

✓ Mathematica. Time used: 2.481 (sec). Leaf size: 71

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

\begin{align*} y(x)\to \frac {c c_1-\sqrt {b^2-c^2 (x-c_1){}^2}}{a} \\ y(x)\to \frac {\sqrt {b^2-c^2 (x-c_1){}^2}+c c_1}{a} \\ \end{align*}

✗ Sympy

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

Timed Out