1.25 problem 3(a)

Internal problem ID [6129]
Internal file name [OUTPUT/5377_Sunday_June_05_2022_03_35_44_PM_61552955/index.tex]

Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS. Page 9
Problem number: 3(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 (1\right ) = 3] \end {align*}

1.25.1 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

1.25.2 Solving as quadrature ode

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

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=1\) and \(y=3\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 3 = c_{1} \end {align*}

The solutions are \begin {align*} c_{1} = 3 \end {align*}

Trying the constant \begin {align*} c_{1} = 3 \end {align*}

Substituting this in the general solution gives \begin {align*} y&=x \,{\mathrm e}^{x}-{\mathrm e}^{x}+3 \end {align*}

The constant \(c_{1} = 3\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= x \,{\mathrm e}^{x}-{\mathrm e}^{x}+3 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = x \,{\mathrm e}^{x}-{\mathrm e}^{x}+3 \] Verified OK.

1.25.3 Maple step by step solution

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

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: 12

dsolve([diff(y(x),x)=x*exp(x),y(1) = 3],y(x), singsol=all)
 

\[ y \left (x \right ) = \left (x -1\right ) {\mathrm e}^{x}+3 \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 14

DSolve[{y'[x]==x*Exp[x],{y[1]==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x (x-1)+3 \]