85.8.4 problem 4

Internal problem ID [22474]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. C Exercises at page 33
Problem number : 4
Date solved : Thursday, October 02, 2025 at 08:40:30 PM
CAS classification : [[_homogeneous, `class G`], _Clairaut]

\begin{align*} y^{\prime }&=\frac {\left (\sqrt {y x +1}-1\right )^{2}}{x^{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.171 (sec). Leaf size: 22

ode=D[y[x],{x,1}]==( (Sqrt[1+x*y[x]] -1)/x)^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {4 (x-c_1)}{c_1{}^2}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.193 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (sqrt(x*y(x) + 1) - 1)**2/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \left (C_{1} x + 2\right ) \]