Internal problem ID [14194]
Internal file name [OUTPUT/13875_Saturday_March_09_2024_03_56_34_PM_12496488/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: 37.
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 {\sqrt {-\left (-1+{\mathrm e}^{2 t +2 c_{1}}\right ) {\mathrm e}^{2 t +2 c_{1}}}}{-1+{\mathrm e}^{2 t +2 c_{1}}}\\ &=\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\mathrm e}^{2 t} c_{1}^{2}}}{-1+{\mathrm e}^{2 t} c_{1}^{2}}\\ y_2&=-\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t +2 c_{1}}\right ) {\mathrm e}^{2 t +2 c_{1}}}}{-1+{\mathrm e}^{2 t +2 c_{1}}}\\ &=-\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\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 {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\mathrm e}^{2 t} c_{1}^{2}}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \\ \tag{2} y &= -\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\mathrm e}^{2 t} c_{1}^{2}}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\mathrm e}^{2 t} c_{1}^{2}}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \] Verified OK.
\[ y = -\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t} c_{1}^{2}\right ) {\mathrm e}^{2 t} c_{1}^{2}}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{3}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{3}+y \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{3}+y}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{3}+y}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (y\right )-\frac {\ln \left (1+y^{2}\right )}{2}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t +2 c_{1}}\right ) {\mathrm e}^{2 t +2 c_{1}}}}{-1+{\mathrm e}^{2 t +2 c_{1}}}, y=-\frac {\sqrt {-\left (-1+{\mathrm e}^{2 t +2 c_{1}}\right ) {\mathrm e}^{2 t +2 c_{1}}}}{-1+{\mathrm e}^{2 t +2 c_{1}}}\right \} \end {array} \]
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 {{\mathrm e}^{-2 t} c_{1} -1}} \\ y \left (t \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{-2 t} c_{1} -1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.048 (sec). Leaf size: 57
DSolve[y'[t]==y[t]^3+y[t],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to -\frac {i e^{t+c_1}}{\sqrt {-1+e^{2 (t+c_1)}}} \\ y(t)\to \frac {i e^{t+c_1}}{\sqrt {-1+e^{2 (t+c_1)}}} \\ \end{align*}