20.4.35 problem Problem 51

Internal problem ID [3670]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.8, Change of Variables. page 79
Problem number : Problem 51
Date solved : Sunday, March 30, 2025 at 02:03:51 AM
CAS classification : [_rational, _Bernoulli]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.225 (sec). Leaf size: 24

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

\[ y(x)\to -\frac {2}{\left (x^2+1\right ) \left (\log \left (x^2+1\right )-2\right )} \]

✓ Sympy. Time used: 0.308 (sec). Leaf size: 27

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

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