44.6.39 problem 39

Internal problem ID [7183]
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 : 39
Date solved : Sunday, March 30, 2025 at 11:50:27 AM
CAS classification : [_linear]

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

With initial conditions

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.089 (sec). Leaf size: 53

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

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

✗ Sympy

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