89.1.25 problem 25

Internal problem ID [24260]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:02:17 PM
CAS classification : [_separable]

\begin{align*} x y y^{\prime }-y^{2}&=1 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 15

ode:=x*y(x)*diff(y(x),x)-y(x)^2 = 1; 
ic:=[y(2) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {2 x^{2}-4}}{2} \]

✓ Mathematica. Time used: 0.248 (sec). Leaf size: 20

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

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

✓ Sympy. Time used: 0.324 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \sqrt {\frac {x^{2}}{2} - 1} \]