2.1 problem 1

Internal problem ID [4749]
Internal file name [OUTPUT/4242_Sunday_June_05_2022_12_46_19_PM_96788878/index.tex]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary differential equations. Section 2. Separable equations. page 398
Problem number: 1.
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]

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

2.1.1 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) &=-\frac {1}{x}\\ q(x) &=0 \end {align*}

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

The domain of \(p(x)=-\frac {1}{x}\) is \[ \{x <0\boldsymbol {\lor }0

2.1.2 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(x,y)\\ &= f( x) g(y)\\ &= \frac {y}{x} \end {align*}

Where \(f(x)=\frac {1}{x}\) and \(g(y)=y\). Integrating both sides gives \begin {align*} \frac {1}{y} \,dy &= \frac {1}{x} \,d x\\ \int { \frac {1}{y} \,dy} &= \int {\frac {1}{x} \,d x}\\ \ln \left (y \right )&=\ln \left (x \right )+c_{1}\\ y&={\mathrm e}^{\ln \left (x \right )+c_{1}}\\ &=c_{1} x \end {align*}

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

The solutions are \begin {align*} c_{1} = {\frac {3}{2}} \end {align*}

Trying the constant \begin {align*} c_{1} = {\frac {3}{2}} \end {align*}

Substituting this in the general solution gives \begin {align*} y&=\frac {3 x}{2} \end {align*}

The constant \(c_{1} = {\frac {3}{2}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {3 x}{2} \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = \frac {3 x}{2} \] Verified OK.

2.1.3 Maple step by step solution

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

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: 0.0 (sec). Leaf size: 7

dsolve([x*diff(y(x),x)=y(x),y(2) = 3],y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {3 x}{2} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 10

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

\[ y(x)\to \frac {3 x}{2} \]