25.6.19 problem 19

Internal problem ID [6903]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 19
Date solved : Tuesday, September 30, 2025 at 04:03:16 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.362 (sec). Leaf size: 49

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

\[ y = \frac {c_1 \sqrt {-\frac {1}{\left (x -c_1 +1\right ) \left (x -c_1 -1\right )}}+1}{\sqrt {-\frac {1}{\left (x -c_1 +1\right ) \left (x -c_1 -1\right )}}} \]

✓ Mathematica. Time used: 29.644 (sec). Leaf size: 20847

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

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✓ Sympy. Time used: 15.086 (sec). Leaf size: 112

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

\[ \left [ y{\left (x \right )} = C_{1} - x + \int \limits ^{- C_{2} + x} \frac {r^{2}}{r^{2} + r \sqrt {2 - r^{2}} - 2}\, dr - \int \limits ^{- C_{2} + x} \frac {r \sqrt {2 - r^{2}}}{r^{2} + r \sqrt {2 - r^{2}} - 2}\, dr, \ y{\left (x \right )} = C_{1} - x + \int \limits ^{- C_{2} + x} \frac {r^{2}}{r^{2} - r \sqrt {2 - r^{2}} - 2}\, dr + \int \limits ^{- C_{2} + x} \frac {r \sqrt {2 - r^{2}}}{r^{2} - r \sqrt {2 - r^{2}} - 2}\, dr\right ] \]