44.6.40 problem 40

Internal problem ID [7184]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.3 Linear equations. Exercises 2.3 at page 63
Problem number : 40
Date solved : Sunday, March 30, 2025 at 11:50:29 AM
CAS classification : [_linear]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }+2 x y&=\left \{\begin {array}{cc} x & 0\le x <1 \\ -x & 1\le x \end {array}\right . \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.062 (sec). Leaf size: 35

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

\[ y = \frac {\left \{\begin {array}{cc} 0 & x <0 \\ x^{2} & x <1 \\ -x^{2}+2 & 1\le x \end {array}\right .}{2 x^{2}+2} \]

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 49

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

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} 0 & x\leq 0 \\ \frac {x^2}{2 x^2+2} & 0<x\leq 1 \\ \frac {2-x^2}{2 x^2+2} & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

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