Internal problem ID [7284]
Internal file name [OUTPUT/6270_Sunday_June_05_2022_04_36_36_PM_75428236/index.tex]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 60.
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]
\[ \boxed {y^{\prime }-y \left (1-y^{2}\right )=0} \]
Integrating both sides gives \begin {align*} \int -\frac {1}{y \left (y^{2}-1\right )}d y &= x +c_{1}\\ \ln \left (y \right )-\frac {\ln \left (y^{2}-1\right )}{2}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\sqrt {\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\\ &=\frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1}\\ y_2&=-\frac {\sqrt {\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\\ &=-\frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \\ \tag{2} y &= -\frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK.
\[ y = -\frac {\sqrt {\left ({\mathrm e}^{2 x} c_{1}^{2}-1\right ) {\mathrm e}^{2 x} c_{1}^{2}}}{{\mathrm e}^{2 x} c_{1}^{2}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \left (1-y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \left (1-y^{2}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \left (1-y^{2}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y \left (1-y^{2}\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (-1+y\right )}{2}+\ln \left (y\right )-\frac {\ln \left (y+1\right )}{2}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}, y=-\frac {\sqrt {\left ({\mathrm e}^{2 x +2 c_{1}}-1\right ) {\mathrm e}^{2 x +2 c_{1}}}}{{\mathrm e}^{2 x +2 c_{1}}-1}\right \} \end {array} \]
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(x),x)=y(x)*(1-y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} +1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{-2 x} c_{1} +1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.787 (sec). Leaf size: 100
DSolve[y'[x]==y[x]*(1-y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {e^x}{\sqrt {e^{2 x}+e^{2 c_1}}} \\ y(x)\to \frac {e^x}{\sqrt {e^{2 x}+e^{2 c_1}}} \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ y(x)\to -\frac {e^x}{\sqrt {e^{2 x}}} \\ y(x)\to \frac {e^x}{\sqrt {e^{2 x}}} \\ \end{align*}