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 : Tuesday, February 04, 2025 at 12:45:14 AM
CAS classification : [_linear]

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

✓ Solution by Maple

Time used: 0.190 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

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

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

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