Internal problem ID [3344]
Internal file name [OUTPUT/2836_Sunday_June_05_2022_08_41_24_AM_4320842/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 3
Problem number: 80.
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]
\[ \boxed {y^{\prime }-y \left (a +b y^{2}\right )=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y \left (b \,y^{2}+a \right )}d y &= x +c_{1}\\ \frac {2 \ln \left (y \right )-\ln \left (b \,y^{2}+a \right )}{2 a}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\frac {\sqrt {-\left (b \,{\mathrm e}^{2 a c_{1} +2 a x}-1\right ) a \,{\mathrm e}^{2 a c_{1} +2 a x}}}{b \,{\mathrm e}^{2 a c_{1} +2 a x}-1}\\ &=\frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1}\\ y_2&=-\frac {\sqrt {-\left (b \,{\mathrm e}^{2 a c_{1} +2 a x}-1\right ) a \,{\mathrm e}^{2 a c_{1} +2 a x}}}{b \,{\mathrm e}^{2 a c_{1} +2 a x}-1}\\ &=-\frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1} \\ \tag{2} y &= -\frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1} \] Verified OK.
\[ y = -\frac {\sqrt {-\left (b \,c_{1}^{2} {\mathrm e}^{2 a x}-1\right ) a \,c_{1}^{2} {\mathrm e}^{2 a x}}}{b \,c_{1}^{2} {\mathrm e}^{2 a x}-1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \left (a +b y^{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \left (a +b y^{2}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \left (a +b y^{2}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y \left (a +b y^{2}\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\ln \left (a +b y^{2}\right )}{2 a}+\frac {\ln \left (y\right )}{a}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & \left \{y=\frac {\sqrt {-\left (b \,{\mathrm e}^{2 a c_{1} +2 a x}-1\right ) a \,{\mathrm e}^{2 a c_{1} +2 a x}}}{b \,{\mathrm e}^{2 a c_{1} +2 a x}-1}, y=-\frac {\sqrt {-\left (b \,{\mathrm e}^{2 a c_{1} +2 a x}-1\right ) a \,{\mathrm e}^{2 a c_{1} +2 a x}}}{b \,{\mathrm e}^{2 a c_{1} +2 a x}-1}\right \} \end {array} \]
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: 0.016 (sec). Leaf size: 70
dsolve(diff(y(x),x) = y(x)*(a+b*y(x)^2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (c_{1} a \,{\mathrm e}^{-2 a x}-b \right ) a}}{c_{1} a \,{\mathrm e}^{-2 a x}-b} \\ y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} a \,{\mathrm e}^{-2 a x}-b \right ) a}}{c_{1} a \,{\mathrm e}^{-2 a x}-b} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.914 (sec). Leaf size: 118
DSolve[y'[x]==y[x](a+b y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \sqrt {a} e^{a (x+c_1)}}{\sqrt {-1+b e^{2 a (x+c_1)}}} \\ y(x)\to \frac {i \sqrt {a} e^{a (x+c_1)}}{\sqrt {-1+b e^{2 a (x+c_1)}}} \\ y(x)\to 0 \\ y(x)\to -\frac {i \sqrt {a}}{\sqrt {b}} \\ y(x)\to \frac {i \sqrt {a}}{\sqrt {b}} \\ \end{align*}