Internal problem ID [12914]
Internal file name [OUTPUT/11567_Tuesday_November_07_2023_11_27_11_PM_31375570/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: 16 (i).
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 (1+y \right )&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {{\mathrm e}^{t +c_{1}}}{-1+{\mathrm e}^{t +c_{1}}}\\ &=-\frac {{\mathrm e}^{t} c_{1}}{-1+c_{1} {\mathrm e}^{t}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {{\mathrm e}^{t} c_{1}}{-1+c_{1} {\mathrm e}^{t}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {{\mathrm e}^{t} c_{1}}{-1+c_{1} {\mathrm e}^{t}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}+y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2}+y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )-\ln \left (y+1\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{t +c_{1}}}{-1+{\mathrm e}^{t +c_{1}}} \end {array} \]
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: 0.016 (sec). Leaf size: 14
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: 0.384 (sec). Leaf size: 33
DSolve[y'[t]==y[t]^2+y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {e^{t+c_1}}{-1+e^{t+c_1}} \\ y(t)\to -1 \\ y(t)\to 0 \\ \end{align*}