85.14.7 problem 5 (b)
Internal
problem
ID
[22523]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
two.
First
order
and
simple
higher
order
ordinary
differential
equations.
C
Exercises
at
page
41
Problem
number
:
5
(b)
Date
solved
:
Thursday, October 02, 2025 at 08:45:36 PM
CAS
classification
:
[[_homogeneous, `class C`], _rational, _dAlembert]
\begin{align*} \sqrt {x +y+1}\, y^{\prime }&=\sqrt {x +y-1} \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 104
ode:=(x+y(x)+1)^(1/2)*diff(y(x),x) = (x+y(x)-1)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
-c_1 +x -y-\frac {y^{2}}{2}-y x -\frac {x^{2}}{2}+\frac {\sqrt {x +y+1}\, \left (x +y-1\right )^{{3}/{2}}}{2}+\frac {\sqrt {x +y+1}\, \sqrt {x +y-1}}{2}-\frac {\sqrt {\left (x +y-1\right ) \left (x +y+1\right )}\, \ln \left (y+x +\sqrt {\left (y+x \right )^{2}-1}\right )}{2 \sqrt {x +y-1}\, \sqrt {x +y+1}} = 0
\]
✓ Mathematica. Time used: 1.217 (sec). Leaf size: 379
ode=Sqrt[x+y[x]+1]*D[y[x],x]== Sqrt[x+y[x]-1];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {1}{2} \sqrt {\frac {1}{-2 y(x)-2 x+2}} \left (2 \sqrt {\frac {y(x)+x-1}{y(x)+x+1}} \sqrt {y(x)+x+1} \left (\text {arctanh}\left (\sqrt {1-\frac {2}{y(x)+x+1}}\right )+\frac {1}{1-\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}}+\frac {1}{\left (\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+1\right )^2}\right )+\sqrt {y(x)+x-1} \sqrt {\frac {y(x)+x-1}{y(x)+x+1}} \left (x^2+2 x y(x)+y(x)^2+y(x)+x\right )-\sqrt {y(x)+x-1} y(x)^2-2 (x-1) \sqrt {y(x)+x-1} y(x)-\sqrt {y(x)+x-1} \log \left (x \sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+y(x)+y(x) \sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+x\right )\right )-\frac {\left (-\frac {1}{y(x)+x-1}\right )^{5/2} (y(x)+x-1)^{5/2} \left (-\text {arctanh}\left (\sqrt {1-\frac {2}{y(x)+x+1}}\right )+\frac {1}{1-\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}}+\frac {2}{\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+1}-\frac {1}{\left (\sqrt {\frac {y(x)+x-1}{y(x)+x+1}}+1\right )^2}\right )}{\sqrt {2}}=c_1,y(x)\right ]
\]
✓ Sympy. Time used: 9.001 (sec). Leaf size: 61
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sqrt(x + y(x) - 1) + sqrt(x + y(x) + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + x + \int \limits ^{- C_{2} - x} \frac {\sqrt {1 - r}}{\sqrt {1 - r} + \sqrt {- r - 1}}\, dr - \int \limits ^{- C_{2} - x} \frac {\sqrt {- r - 1}}{\sqrt {1 - r} + \sqrt {- r - 1}}\, dr
\]