29.37.5 problem 1118
Internal
problem
ID
[5663]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
37
Problem
number
:
1118
Date
solved
:
Tuesday, March 04, 2025 at 11:17:23 PM
CAS
classification
:
[_quadrature]
\begin{align*} \sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }&=x \end{align*}
✓ Maple. Time used: 0.088 (sec). Leaf size: 113
ode:=(1+diff(y(x),x)^2)^(1/2)+a*diff(y(x),x) = x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \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} \\
y \left (x \right ) &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_{1} \right ) \left (a -1\right ) \left (a +1\right )}{2 a^{2}-2} \\
\end{align*}
✓ Mathematica. Time used: 0.225 (sec). Leaf size: 154
ode=Sqrt[1+(D[y[x],x])^2]+ a*D[y[x],x]==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\left (a^2-1\right )^{3/2} \left (-\sqrt {\frac {a^2+x^2-1}{a^2-1}}\right ) \text {arcsinh}\left (\frac {x}{\sqrt {a^2-1}}\right )-x \left (-a x \sqrt {a^2+x^2-1}+a^2+x^2-1\right )}{2 \left (a^2-1\right ) \sqrt {a^2+x^2-1}}+c_1 \\
y(x)\to \frac {1}{2} \left (\frac {x \left (\sqrt {a^2+x^2-1}+a x\right )}{a^2-1}+\log \left (\sqrt {a^2+x^2-1}+x\right )\right )+c_1 \\
\end{align*}
✓ Sympy. Time used: 2.184 (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)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {- a x^{2} + x \sqrt {a^{2} + x^{2} - 1} + \left (a^{2} - 1\right ) \left (\begin {cases} \log {\left (2 x + 2 \sqrt {a^{2} + x^{2} - 1} \right )} & \text {for}\: a^{2} \neq 1 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right )}{2 \left (a^{2} - 1\right )}, \ y{\left (x \right )} = C_{1} - \frac {- a x^{2} - x \sqrt {a^{2} + x^{2} - 1} - \left (a^{2} - 1\right ) \left (\begin {cases} \log {\left (2 x + 2 \sqrt {a^{2} + x^{2} - 1} \right )} & \text {for}\: a^{2} \neq 1 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2}}} & \text {otherwise} \end {cases}\right )}{2 \left (a^{2} - 1\right )}\right ]
\]