6.2.9 problem 9

Internal problem ID [1545]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Linear first order. Section 2.1 Page 41
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:35:49 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\frac {2 x y}{x^{2}+1}&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 11

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

\[ y = 2 x^{2}+2 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 12

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

\begin{align*} y(x)&\to 2 \left (x^2+1\right ) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 8

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

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