58.4.14 problem 14

Internal problem ID [14584]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:43:04 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} \sqrt {x +y}+\sqrt {x -y}+\left (\sqrt {x -y}-\sqrt {x +y}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 2.018 (sec). Leaf size: 36

ode:=(x+y(x))^(1/2)+(x-y(x))^(1/2)+((x-y(x))^(1/2)-(x+y(x))^(1/2))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.838 (sec). Leaf size: 52

ode=(Sqrt[x+y[x]]+Sqrt[x-y[x]])+(Sqrt[x-y[x]]-Sqrt[x+y[x]])*D[y[x],x]==0; 
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: 5.921 (sec). Leaf size: 92

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

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