Internal problem ID [14225]
Internal file name [OUTPUT/13906_Saturday_March_09_2024_03_57_52_PM_12423894/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number: 65.
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]
\[ \boxed {y^{\prime }-y^{2}+y=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}-y}d y &= t +c_{1}\\ -\ln \left (y \right )+\ln \left (y -1\right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {1}{-1+{\mathrm e}^{t +c_{1}}}\\ &=-\frac {1}{-1+c_{1} {\mathrm e}^{t}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {1}{-1+c_{1} {\mathrm e}^{t}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {1}{-1+c_{1} {\mathrm e}^{t}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}-y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2}-y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y-1\right )-\ln \left (y\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {1}{-1+{\mathrm e}^{t +c_{1}}} \end {array} \]
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: 0.0 (sec). Leaf size: 12
dsolve(diff(y(t),t)=y(t)^2-y(t),y(t), singsol=all)
\[ y \left (t \right ) = \frac {1}{1+{\mathrm e}^{t} c_{1}} \]
✓ Solution by Mathematica
Time used: 0.194 (sec). Leaf size: 25
DSolve[y'[t]==y[t]^2-y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{1+e^{t+c_1}} \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}