4.6 problem 7

Internal problem ID [537]
Internal file name [OUTPUT/537_Sunday_June_05_2022_01_43_52_AM_26451284/index.tex]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Section 2.5. Page 88
Problem number: 7.
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]

\[ \boxed {y^{\prime }+k \left (-1+y\right )^{2}=0} \]

4.6.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int -\frac {1}{k \left (y -1\right )^{2}}d y &= t +c_{1}\\ \frac {1}{\left (y -1\right ) k}&=t +c_{1} \end {align*}

Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {c_{1} k +t k +1}{k \left (t +c_{1} \right )} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} k +t k +1}{k \left (t +c_{1} \right )} \\ \end{align*}

Verification of solutions

\[ y = \frac {c_{1} k +t k +1}{k \left (t +c_{1} \right )} \] Verified OK.

4.6.2 Maple step by step solution

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

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: 20

dsolve(diff(y(t),t) = -k*(-1+y(t))^2,y(t), singsol=all)
 

\[ y \left (t \right ) = \frac {1+k \left (t +c_{1} \right )}{k \left (t +c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.139 (sec). Leaf size: 30

DSolve[y'[t]== -k*(-1+y[t])^2,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to \frac {k t+1-c_1}{k t-c_1} \\ y(t)\to 1 \\ \end{align*}