21.6 problem 2(a)

Internal problem ID [6068]
Internal file name [OUTPUT/5316_Sunday_June_05_2022_03_33_58_PM_44743446/index.tex]

Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section: Chapter 5. Existence and uniqueness of solutions to first order equations. Page 190
Problem number: 2(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]

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

21.6.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(x,y)\\ &= y^{2} \end {align*}

The \(y\) domain of \(f(x,y)\) when \(x=x_{0}\) is \[ \{-\infty

21.6.2 Solving as quadrature ode

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

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

The solutions are \begin {align*} c_{1} = -\frac {x_{0} y_{0} +1}{y_{0}} \end {align*}

Trying the constant \begin {align*} c_{1} = -\frac {x_{0} y_{0} +1}{y_{0}} \end {align*}

Substituting \(c_{1}\) found above in the general solution gives \begin {align*} -\frac {1}{y} = x -\frac {x_{0} y_{0} +1}{y_{0}} \end {align*}

The constant \(c_{1} = -\frac {x_{0} y_{0} +1}{y_{0}}\) gives valid solution.

Summary

The solution(s) found are the following \begin{align*} \tag{1} -\frac {1}{y} &= \frac {-1+\left (x -x_{0} \right ) y_{0}}{y_{0}} \\ \end{align*}

Verification of solutions

\[ -\frac {1}{y} = \frac {-1+\left (x -x_{0} \right ) y_{0}}{y_{0}} \] Verified OK.

21.6.3 Maple step by step solution

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

Maple trace

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 18

dsolve([diff(y(x),x)=y(x)^2,y(x__0) = y__0],y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {y_{0}}{-1+\left (x -x_{0} \right ) y_{0}} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 16

DSolve[{y'[x]==x2*y[x],{y[x0]==y0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {y0} e^{\text {x2} (x-\text {x0})} \]