Internal problem ID [13042]
Internal file name [OUTPUT/11695_Wednesday_November_08_2023_03_28_53_AM_26392571/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. Review Exercises for chapter 1. page
136
Problem number: 28.
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]
\[ \boxed {y^{\prime }-2 y+y^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+2 y}d y &= t +c_{1}\\ -\frac {\ln \left (y -2\right )}{2}+\frac {\ln \left (y \right )}{2}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {2 \,{\mathrm e}^{2 t +2 c_{1}}}{-1+{\mathrm e}^{2 t +2 c_{1}}}\\ &=\frac {2 \,{\mathrm e}^{2 t} c_{1}^{2}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {2 \,{\mathrm e}^{2 t} c_{1}^{2}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {2 \,{\mathrm e}^{2 t} c_{1}^{2}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-2 y+y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=2 y-y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{2 y-y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{2 y-y^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (y-2\right )}{2}+\frac {\ln \left (y\right )}{2}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 \,{\mathrm e}^{2 t +2 c_{1}}}{-1+{\mathrm e}^{2 t +2 c_{1}}} \end {array} \]
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: 17
dsolve(diff(y(t),t)= 2*y(t)-y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {2}{1+2 \,{\mathrm e}^{-2 t} c_{1}} \]
✓ Solution by Mathematica
Time used: 0.447 (sec). Leaf size: 36
DSolve[y'[t]==2*y[t]-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {2 e^{2 t}}{e^{2 t}+e^{2 c_1}} \\ y(t)\to 0 \\ y(t)\to 2 \\ \end{align*}