Internal problem ID [14227]
Internal file name [OUTPUT/13908_Saturday_March_09_2024_03_57_52_PM_45933005/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: 67.
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]
\[ \boxed {y^{\prime }-4 y+y^{2}=12} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}+4 y +12}d y &= t +c_{1}\\ -\frac {\ln \left (y -6\right )}{8}+\frac {\ln \left (y +2\right )}{8}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {6 \,{\mathrm e}^{8 t +8 c_{1}}+2}{{\mathrm e}^{8 t +8 c_{1}}-1}\\ &=\frac {6 \,{\mathrm e}^{8 t} c_{1}^{8}+2}{{\mathrm e}^{8 t} c_{1}^{8}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {6 \,{\mathrm e}^{8 t} c_{1}^{8}+2}{{\mathrm e}^{8 t} c_{1}^{8}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {6 \,{\mathrm e}^{8 t} c_{1}^{8}+2}{{\mathrm e}^{8 t} c_{1}^{8}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-4 y+y^{2}=12 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=12+4 y-y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{12+4 y-y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{12+4 y-y^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (y-6\right )}{8}+\frac {\ln \left (y+2\right )}{8}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {2 \left (3 \,{\mathrm e}^{8 t +8 c_{1}}+1\right )}{{\mathrm e}^{8 t +8 c_{1}}-1} \end {array} \]
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: 0.047 (sec). Leaf size: 24
dsolve(diff(y(t),t)=12+4*y(t)-y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \frac {6 c_{1} {\mathrm e}^{8 t}+2}{c_{1} {\mathrm e}^{8 t}-1} \]
✓ Solution by Mathematica
Time used: 0.463 (sec). Leaf size: 47
DSolve[y'[t]==12+4*y[t]-y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {2 \left (3 e^{8 t}+e^{8 c_1}\right )}{e^{8 t}-e^{8 c_1}} \\ y(t)\to -2 \\ y(t)\to 6 \\ \end{align*}