Internal problem ID [14196]
Internal file name [OUTPUT/13877_Saturday_March_09_2024_03_56_35_PM_50430242/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: 39.
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]
\[ \boxed {y^{\prime }-y^{3}+y=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{3}-y}d y &= t +c_{1}\\ -\ln \left (y \right )+\frac {\ln \left (y^{2}-1\right )}{2}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {1}{\sqrt {1-{\mathrm e}^{2 t +2 c_{1}}}}\\ &=\frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}}\\ y_2&=-\frac {1}{\sqrt {1-{\mathrm e}^{2 t +2 c_{1}}}}\\ &=-\frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}} \\ \tag{2} y &= -\frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}} \\ \end{align*}
Verification of solutions
\[ y = \frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}} \] Verified OK.
\[ y = -\frac {1}{\sqrt {1-{\mathrm e}^{2 t} c_{1}^{2}}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{3}+y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{3}-y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}-y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{3}-y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y-1\right )}{2}-\ln \left (y\right )+\frac {\ln \left (y+1\right )}{2}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {1}{\sqrt {1-{\mathrm e}^{2 t +2 c_{1}}}}, y=-\frac {1}{\sqrt {1-{\mathrm e}^{2 t +2 c_{1}}}}\right \} \end {array} \]
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: 29
dsolve(diff(y(t),t)=y(t)^3-y(t),y(t), singsol=all)
\begin{align*} y \left (t \right ) &= \frac {1}{\sqrt {c_{1} {\mathrm e}^{2 t}+1}} \\ y \left (t \right ) &= -\frac {1}{\sqrt {c_{1} {\mathrm e}^{2 t}+1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.714 (sec). Leaf size: 54
DSolve[y'[t]==y[t]^3-y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {1}{\sqrt {1+e^{2 (t+c_1)}}} \\ y(t)\to \frac {1}{\sqrt {1+e^{2 (t+c_1)}}} \\ y(t)\to -1 \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}