58.6.19 problem 19

Internal problem ID [14650]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, Miscellaneous Review. Exercises page 60
Problem number : 19
Date solved : Thursday, October 02, 2025 at 09:46:06 AM
CAS classification : [_separable]

\begin{align*} 4 x y y^{\prime }&=1+y^{2} \end{align*}

With initial conditions

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

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

ode:=4*x*y(x)*diff(y(x),x) = 1+y(x)^2; 
ic:=[y(2) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sqrt {\sqrt {x}\, \sqrt {2}-1} \]

✓ Mathematica. Time used: 3.416 (sec). Leaf size: 22

ode=4*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 \sqrt {\sqrt {2} \sqrt {x}-1} \end{align*}

✓ Sympy. Time used: 0.329 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*y(x)*Derivative(y(x), x) - y(x)**2 - 1,0) 
ics = {y(2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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