29.14.10 problem 391
Internal
problem
ID
[4989]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
14
Problem
number
:
391
Date
solved
:
Sunday, March 30, 2025 at 04:28:36 AM
CAS
classification
:
[_separable]
\begin{align*} \left (x -\sqrt {x^{2}+1}\right ) y^{\prime }&=y+\sqrt {1+y^{2}} \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 39
ode:=(x-(x^2+1)^(1/2))*diff(y(x),x) = y(x)+(1+y(x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
c_1 +x^{2}+x \sqrt {x^{2}+1}+\operatorname {arcsinh}\left (x \right )+y \sqrt {1+y^{2}}+\operatorname {arcsinh}\left (y\right )-y^{2} = 0
\]
✓ Mathematica. Time used: 0.883 (sec). Leaf size: 84
ode=(x-Sqrt[1+x^2])D[y[x],x]==y[x]+Sqrt[1+ y[x]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )\right )\&\right ]\left [\frac {1}{2} \left (\log \left (\sqrt {x^2+1}-x\right )-x \left (\sqrt {x^2+1}+x\right )\right )+c_1\right ]
\]
✓ Sympy. Time used: 1.666 (sec). Leaf size: 138
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x - sqrt(x**2 + 1))*Derivative(y(x), x) - sqrt(y(x)**2 + 1) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\frac {x \operatorname {asinh}{\left (x \right )}}{2 \left (x - \sqrt {x^{2} + 1}\right )} - \frac {x}{2 \left (x - \sqrt {x^{2} + 1}\right )} + \frac {\sqrt {y^{2}{\left (x \right )} + 1} \operatorname {asinh}{\left (y{\left (x \right )} \right )}}{2 \left (\sqrt {y^{2}{\left (x \right )} + 1} + y{\left (x \right )}\right )} + \frac {y{\left (x \right )} \operatorname {asinh}{\left (y{\left (x \right )} \right )}}{2 \left (\sqrt {y^{2}{\left (x \right )} + 1} + y{\left (x \right )}\right )} + \frac {y{\left (x \right )}}{2 \left (\sqrt {y^{2}{\left (x \right )} + 1} + y{\left (x \right )}\right )} - \frac {\sqrt {x^{2} + 1} \operatorname {asinh}{\left (x \right )}}{2 \left (x - \sqrt {x^{2} + 1}\right )} = C_{1}
\]