4.2 problem 6

Internal problem ID [12942]
Internal file name [OUTPUT/11595_Tuesday_November_07_2023_11_31_55_PM_94325409/index.tex]

Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.5 page 71
Problem number: 6.
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]

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

4.2.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(t,y)\\ &= y \left (y -1\right ) \left (y -3\right ) \end {align*}

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

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(t=0\) is \[ \{-\infty

4.2.2 Solving as quadrature ode

Since ode has form \(y^{\prime }= f(y)\) and initial conditions \(y = 0\) is verified to satisfy the ode, then the solution is \begin {align*} y&=y_0 \\ &=0 \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= 0 \\ \end{align*}

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ y = 0 \] Verified OK.

Maple trace

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

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 5

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

\[ y \left (t \right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 6

DSolve[{y'[t]==y[t]*(y[t]-1)*(y[t]-3),{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to 0 \]