85.14.1 problem 1 (a)

Internal problem ID [22517]
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 : 1 (a)
Date solved : Thursday, October 02, 2025 at 08:45:12 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=\frac {\sqrt {x +y}+\sqrt {x -y}}{\sqrt {x +y}-\sqrt {x -y}} \end{align*}

✓ Maple. Time used: 2.277 (sec). Leaf size: 32

ode:=diff(y(x),x) = ((x+y(x))^(1/2)+(x-y(x))^(1/2))/((x+y(x))^(1/2)-(x-y(x))^(1/2)); 
dsolve(ode,y(x), singsol=all);
 

\[ \ln \left (x \right )+\ln \left (\frac {y}{x}\right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {y^{2}}{x^{2}}+1}}\right )-c_1 = 0 \]

✓ Mathematica. Time used: 0.839 (sec). Leaf size: 52

ode=D[y[x],x]==  ( Sqrt[x+y[x]] + Sqrt[x-y[x]] )/( Sqrt[x+y[x]] - Sqrt[x-y[x]] ) ; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to \sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 5.318 (sec). Leaf size: 92

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (sqrt(x - y(x)) + sqrt(x + y(x)))/(-sqrt(x - y(x)) + sqrt(x + y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1} - 1}}{u_{1} \sqrt {u_{1} - 1} + u_{1} \sqrt {u_{1} + 1} + \sqrt {u_{1} - 1} - \sqrt {u_{1} + 1}}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1} + 1}}{u_{1} \sqrt {u_{1} - 1} + u_{1} \sqrt {u_{1} + 1} + \sqrt {u_{1} - 1} - \sqrt {u_{1} + 1}}\, du_{1}} \]