76.1.18 problem 18

Internal problem ID [17246]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 18
Date solved : Monday, March 31, 2025 at 03:46:08 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (2\right )&=0 \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 17

ode:=diff(y(x),x) = 2*x/(1+2*y(x)); 
ic:=y(2) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {1}{2}+\frac {\sqrt {4 x^{2}-15}}{2} \]

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 22

ode=D[y[x],x]==2*x/(1+2*y[x]); 
ic={y[2]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} \left (\sqrt {4 x^2-15}-1\right ) \]

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

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

\[ y{\left (x \right )} = \frac {\sqrt {4 x^{2} - 15}}{2} - \frac {1}{2} \]