60.1.489 problem 502

Internal problem ID [10503]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 502
Date solved : Sunday, March 30, 2025 at 05:22:27 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.285 (sec). Leaf size: 200

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

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

✓ Mathematica. Time used: 2.535 (sec). Leaf size: 71

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

\begin{align*} y(x)\to \frac {b c_1-\sqrt {c^2-b^2 (x-c_1){}^2}}{a} \\ y(x)\to \frac {\sqrt {c^2-b^2 (x-c_1){}^2}+b 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(-c**2*(a*Derivative(y(x), x) + b)**2 + (a*y(x) - b*x)**2*(a**2*Derivative(y(x), x)**2 + b**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out