1.9 problem 4(a)

Internal problem ID [877]
Internal file name [OUTPUT/877_Sunday_June_05_2022_01_53_01_AM_12765659/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 1, Introduction. Section 1.2 Page 14
Problem number: 4(a).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{\prime }=-x \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}

1.9.1 Existence and uniqueness analysis

This is a linear ODE. In canonical form it is written as \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}

Where here \begin {align*} p(x) &=0\\ q(x) &=-x \,{\mathrm e}^{x} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = -x \,{\mathrm e}^{x} \end {align*}

The domain of \(p(x)=0\) is \[ \{-\infty

1.9.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { -x \,{\mathrm e}^{x}\,\mathop {\mathrm {d}x}}\\ &= -\left (x -1\right ) {\mathrm e}^{x}+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=1\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 1 = 1+c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = 0 \end {align*}

Trying the constant \begin {align*} c_{1} = 0 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=-x \,{\mathrm e}^{x}+{\mathrm e}^{x} \end {align*}

The constant \(c_{1} = 0\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -x \,{\mathrm e}^{x}+{\mathrm e}^{x} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = -x \,{\mathrm e}^{x}+{\mathrm e}^{x} \] Verified OK.

1.9.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=-x \,{\mathrm e}^{x}, y \left (0\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int -x \,{\mathrm e}^{x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\left (x -1\right ) {\mathrm e}^{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-x \,{\mathrm e}^{x}+{\mathrm e}^{x}+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=1 \\ {} & {} & 1=1+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{x} \left (1-x \right ) \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y={\mathrm e}^{x} \left (1-x \right ) \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 11

dsolve([diff(y(x),x) = -x*exp(x),y(0) = 1],y(x), singsol=all)
 

\[ y \left (x \right ) = -\left (x -1\right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 13

DSolve[{y'[x] == -x*Exp[x],y[0]==1},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -e^x (x-1) \]