2.13 problem 15 b(3)

Internal problem ID [12911]
Internal file name [OUTPUT/11564_Tuesday_November_07_2023_11_27_08_PM_32998932/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 1.3 page 47
Problem number: 15 b(3).
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]

\[ \boxed {S^{\prime }-S^{3}+2 S^{2}-S=0} \] With initial conditions \begin {align*} [S \left (0\right ) = 1] \end {align*}

2.13.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} S^{\prime } &= f(t,S)\\ &= S^{3}-2 S^{2}+S \end {align*}

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

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

2.13.2 Solving as quadrature ode

Since ode has form \(S^{\prime }= f(S)\) and initial conditions \(S = 1\) is verified to satisfy the ode, then the solution is \begin {align*} S&=S_0 \\ &=1 \end {align*}

Summary

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

(a) Solution plot

(b) Slope field plot

Verification of solutions

\[ S = 1 \] Verified OK.

2.13.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [S^{\prime }-S^{3}+2 S^{2}-S=0, S \left (0\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & S^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & S^{\prime }=S^{3}-2 S^{2}+S \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {S^{\prime }}{S^{3}-2 S^{2}+S}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {S^{\prime }}{S^{3}-2 S^{2}+S}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{S-1}-\ln \left (S-1\right )+\ln \left (S\right )=t +c_{1} \\ \bullet & {} & \textrm {Solution does not satisfy initial condition}\hspace {3pt} \end {array} \]

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(S(t),t)=S(t)^3-2*S(t)^2+S(t),S(0) = 1],S(t), singsol=all)
 

\[ S \left (t \right ) = 1 \]

✓ Solution by Mathematica

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

DSolve[{S'[t]==S[t]^3-2*S[t]^2+S[t],{S[0]==1}},S[t],t,IncludeSingularSolutions -> True]
 

\[ S(t)\to 1 \]