31.6.7 problem 7
Internal
problem
ID
[5756]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
7
Date
solved
:
Sunday, March 30, 2025 at 10:08:14 AM
CAS
classification
:
[_quadrature]
\begin{align*} y&=a y^{\prime }+\sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Maple. Time used: 0.068 (sec). Leaf size: 112
ode:=y(x) = (1+diff(y(x),x)^2)^(1/2)+a*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
-\int _{}^{y}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} a^{2}+\int _{}^{y}\frac {1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_1 +x &= 0 \\
\int _{}^{y}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} a^{2}-\int _{}^{y}\frac {1}{-a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_1 +x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.878 (sec). Leaf size: 210
ode=y[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 \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a+1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a-1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\
y(x)\to 1 \\
\end{align*}
✓ Sympy. Time used: 1.587 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*Derivative(y(x), x) - sqrt(Derivative(y(x), x)**2 + 1) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{y a - \sqrt {y^{2} + a^{2} - 1}}\, dy = C_{1} + \frac {x}{\left (a - 1\right ) \left (a + 1\right )}, \ \int \limits ^{y{\left (x \right )}} \frac {1}{y a + \sqrt {y^{2} + a^{2} - 1}}\, dy = C_{1} + \frac {x}{\left (a - 1\right ) \left (a + 1\right )}\right ]
\]