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 : Tuesday, February 04, 2025 at 12:45:05 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }+2 y x&=\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*}

✓ Solution by Maple

Time used: 0.302 (sec). Leaf size: 52

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

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

✓ Solution by Mathematica

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

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

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