3.11 problem 2(a)

Internal problem ID [6167]
Internal file name [OUTPUT/5415_Sunday_June_05_2022_03_36_39_PM_42878629/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 1.4 First Order Linear Equations. Page 15
Problem number: 2(a).
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "exact", "linear", "separable", "homogeneousTypeD2", "first_order_ode_lie_symmetry_lookup"

Maple gives the following as the ode type

[_separable]

\[ \boxed {y^{\prime }-y x=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}

3.11.1 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) &=-x\\ q(x) &=0 \end {align*}

Hence the ode is \begin {align*} y^{\prime }-y x = 0 \end {align*}

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

3.11.2 Solving as linear ode

Entering Linear first order ODE solver. The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -x d x} \\ &= {\mathrm e}^{-\frac {x^{2}}{2}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu y &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left ({\mathrm e}^{-\frac {x^{2}}{2}} y\right ) &= 0 \end {align*}

Integrating gives \begin {align*} {\mathrm e}^{-\frac {x^{2}}{2}} y &= c_{1} \end {align*}

Dividing both sides by the integrating factor \(\mu ={\mathrm e}^{-\frac {x^{2}}{2}}\) results in \begin {align*} y &= c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \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} {\mathrm e}^{\frac {1}{2}} \end {align*}

The solutions are \begin {align*} c_{1} = 3 \,{\mathrm e}^{-\frac {1}{2}} \end {align*}

Trying the constant \begin {align*} c_{1} = 3 \,{\mathrm e}^{-\frac {1}{2}} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=3 \,{\mathrm e}^{\frac {\left (x -1\right ) \left (1+x \right )}{2}} \end {align*}

The constant \(c_{1} = 3 \,{\mathrm e}^{-\frac {1}{2}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 3 \,{\mathrm e}^{\frac {\left (x -1\right ) \left (1+x \right )}{2}} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 3 \,{\mathrm e}^{\frac {\left (x -1\right ) \left (1+x \right )}{2}} \] Verified OK.

3.11.3 Maple step by step solution

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

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`
 

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 15

dsolve([diff(y(x),x)-x*y(x)=0,y(1) = 3],y(x), singsol=all)
 

\[ y \left (x \right ) = 3 \,{\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{2}} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 18

DSolve[{y'[x]-x*y[x]==0,{y[1]==3}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to 3 e^{\frac {1}{2} \left (x^2-1\right )} \]