25.6.11 problem 11

Internal problem ID [6895]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 04:00:25 PM
CAS classification : [_quadrature]

\begin{align*} 1+{y^{\prime }}^{2}&=\frac {\left (x +a \right )^{2}}{2 a x +x^{2}} \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 42

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

\begin{align*} y &= a \ln \left (a +x +\sqrt {x \left (2 a +x \right )}\right )+c_1 \\ y &= -a \ln \left (a +x +\sqrt {x \left (2 a +x \right )}\right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 103

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

\begin{align*} y(x)&\to -\frac {2 a \sqrt {x} \sqrt {2 a+x} \text {arctanh}\left (\frac {\sqrt {2 a+x}}{\sqrt {x}}\right )}{\sqrt {x (2 a+x)}}+c_1\\ y(x)&\to \frac {2 a \sqrt {x} \sqrt {2 a+x} \text {arctanh}\left (\frac {\sqrt {2 a+x}}{\sqrt {x}}\right )}{\sqrt {x (2 a+x)}}+c_1 \end{align*}

✓ Sympy. Time used: 0.780 (sec). Leaf size: 37

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

\[ \left [ y{\left (x \right )} = C_{1} - a \int \sqrt {\frac {1}{x \left (2 a + x\right )}}\, dx, \ y{\left (x \right )} = C_{1} + a \int \sqrt {\frac {1}{x \left (2 a + x\right )}}\, dx\right ] \]