44.6.41 problem 41

Internal problem ID [7185]
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 : 41
Date solved : Tuesday, February 04, 2025 at 12:45:22 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+\left (\left \{\begin {array}{cc} 2 & 0\le x \le 1 \\ -\frac {2}{x} & 1<x \end {array}\right .\right ) y&=4 x \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3 \end{align*}

✓ Solution by Maple

Time used: 0.291 (sec). Leaf size: 46

dsolve([diff(y(x),x)+piecewise(0<=x and x<=1,2,x>1,-2/x)*y(x)=4*x,y(0) = 3],y(x), singsol=all)
 

\[ y = \left \{\begin {array}{cc} 2 x^{2}+3 & x <0 \\ 2 x -1+4 \,{\mathrm e}^{-2 x} & x <1 \\ x^{2} \left (4 \ln \left (x \right )+4 \,{\mathrm e}^{-2}+1\right ) & 1\le x \end {array}\right . \]

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 52

DSolve[{D[y[x],x]+Piecewise[{ {2,0<=x<=1},{-2/x,x>1}}]*y[x]==4*x,{y[0]==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 x^2+3 & x\leq 0 \\ 2 x+4 e^{-2 x}-1 & 0<x\leq 1 \\ x^2 \left (4 \log (x)+\frac {4}{e^2}+1\right ) & \text {True} \\ \end {array} \\ \end {array} \]