38.6.41 problem 41

Internal problem ID [8471]
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, September 30, 2025 at 05:37:41 PM
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*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 46

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

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

✓ Mathematica. Time used: 0.221 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} 2 x^2+3 & x\leq 0 \\ \frac {4 e^2 \log (x) x^2+e^2 x^2+4 x^2}{e^2} & x>1 \\ e^{-2 x} \left (\int _0^x4 e^{2 K[1]} K[1]dK[1]+3\right ) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x + Piecewise((2, (x >= 0) & (x <= 1)), (-2/x, x > 1))*y(x) + Derivative(y(x), x),0) 
ics = {y(0): 3} 
dsolve(ode,func=y(x),ics=ics)