1.37 problem 2.7 a

Internal problem ID [13278]
Internal file name [OUTPUT/12450_Wednesday_February_14_2024_02_06_19_AM_62722038/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page 32
Problem number: 2.7 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]

\[ \boxed {y^{\prime }=x \,{\mathrm e}^{-x^{2}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}

1.37.1 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^{2}} \end {align*}

Hence the ode is \begin {align*} y^{\prime } = x \,{\mathrm e}^{-x^{2}} \end {align*}

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

1.37.2 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { x \,{\mathrm e}^{-x^{2}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {{\mathrm e}^{-x^{2}}}{2}+c_{1} \end {align*}

Initial conditions are used to solve for \(c_{1}\). Substituting \(x=0\) and \(y=3\) in the above solution gives an equation to solve for the constant of integration. \begin {align*} 3 = c_{1} -\frac {1}{2} \end {align*}

The solutions are \begin {align*} c_{1} = {\frac {7}{2}} \end {align*}

Trying the constant \begin {align*} c_{1} = {\frac {7}{2}} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=-\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \end {align*}

The constant \(c_{1} = {\frac {7}{2}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = -\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \] Verified OK.

1.37.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=x \,{\mathrm e}^{-x^{2}}, y \left (0\right )=3\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int x \,{\mathrm e}^{-x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{-x^{2}}}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{-x^{2}}}{2}+c_{1} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=3 \\ {} & {} & 3=c_{1} -\frac {1}{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} c_{1} \\ {} & {} & c_{1} =\frac {7}{2} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} c_{1} =\frac {7}{2}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & y=-\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \end {array} \]

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: 14

dsolve([diff(y(x),x)=x*exp(-x^2),y(0) = 3],y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {{\mathrm e}^{-x^{2}}}{2}+\frac {7}{2} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 20

DSolve[{y'[x]==x*Exp[-x^2],{y[0]==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {7}{2}-\frac {e^{-x^2}}{2} \]