25.6.17 problem 17

Internal problem ID [6901]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 04:02:38 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} x +y y^{\prime }&=a \sqrt {1+{y^{\prime }}^{2}} \end{align*}

✓ Maple. Time used: 0.375 (sec). Leaf size: 237

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 2.204 (sec). Leaf size: 150

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

\begin{align*} \text {Solve}\left [\frac {a^2 \arctan \left (\frac {\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}\right )+a^2 \arctan \left (\frac {x}{y(x)}\right )-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {-a^2 \arctan \left (\frac {\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}\right )+a^2 \arctan \left (\frac {x}{y(x)}\right )+\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}=c_1,y(x)\right ] \end{align*}

✗ Sympy

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

Timed Out