25.6.8 problem 8
Internal
problem
ID
[6892]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
8
Date
solved
:
Tuesday, September 30, 2025 at 04:00:17 PM
CAS
classification
:
[_quadrature]
\begin{align*} x&=a y^{\prime }+\sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Maple. Time used: 0.070 (sec). Leaf size: 113
ode:=x = a*diff(y(x),x)+(1+diff(y(x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {a \,x^{2}+x \sqrt {a^{2}+x^{2}-1}+\left (a -1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )+2 c_1 \right ) \left (a +1\right )}{2 a^{2}-2} \\
y &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (a -1\right ) \left (a +1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_1 \right )}{2 a^{2}-2} \\
\end{align*}
✓ Mathematica. Time used: 0.025 (sec). Leaf size: 109
ode=x==a*D[y[x],x]+Sqrt[1+(D[y[x],x])^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{2} \left (\frac {x \left (a x-\sqrt {a^2+x^2-1}\right )}{a^2-1}-\text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2-1}}\right )\right )+c_1\\ y(x)&\to \frac {1}{2} \left (\text {arctanh}\left (\frac {x}{\sqrt {a^2+x^2-1}}\right )+\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}\right )+c_1 \end{align*}
✓ Sympy. Time used: 1.342 (sec). Leaf size: 134
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*Derivative(y(x), x) + x - sqrt(Derivative(y(x), x)**2 + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {- a x^{2} + x \sqrt {a^{2} + x^{2} - 1} + \left (a^{2} - 1\right ) \left (\begin {cases} \log {\left (2 x + 2 \sqrt {a^{2} + x^{2} - 1} \right )} & \text {for}\: a^{2} \neq 1 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right )}{2 \left (a^{2} - 1\right )}, \ y{\left (x \right )} = C_{1} - \frac {- a x^{2} - x \sqrt {a^{2} + x^{2} - 1} - \left (a^{2} - 1\right ) \left (\begin {cases} \log {\left (2 x + 2 \sqrt {a^{2} + x^{2} - 1} \right )} & \text {for}\: a^{2} \neq 1 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right )}{2 \left (a^{2} - 1\right )}\right ]
\]