Internal problem ID [13295]
Internal file name [OUTPUT/12467_Wednesday_February_14_2024_02_06_29_AM_27016616/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 3. Some basics about First order equations. Additional exercises. page
63
Problem number: 3.4 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]
\[ \boxed {y^{\prime }+2 y-y^{2}=-2} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}-2 y -2}d y &= x +c_{1}\\ -\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\left (2 y -2\right ) \sqrt {3}}{6}\right )}{3}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {\left (-\sqrt {3}+3 \tanh \left (\left (x +c_{1} \right ) \sqrt {3}\right )\right ) \sqrt {3}}{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {\left (-\sqrt {3}+3 \tanh \left (\left (x +c_{1} \right ) \sqrt {3}\right )\right ) \sqrt {3}}{3} \\ \end{align*}
Verification of solutions
\[ y = -\frac {\left (-\sqrt {3}+3 \tanh \left (\left (x +c_{1} \right ) \sqrt {3}\right )\right ) \sqrt {3}}{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 y-y^{2}=-2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-2 y+y^{2}-2 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-2 y+y^{2}-2}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-2 y+y^{2}-2}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {\sqrt {3}\, \mathrm {arctanh}\left (\frac {\left (2 y-2\right ) \sqrt {3}}{6}\right )}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {\left (-\sqrt {3}+3 \tanh \left (\left (x +c_{1} \right ) \sqrt {3}\right )\right ) \sqrt {3}}{3} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(diff(y(x),x)+2*y(x)-y(x)^2=-2,y(x), singsol=all)
\[ y \left (x \right ) = 1-\sqrt {3}\, \tanh \left (\left (c_{1} +x \right ) \sqrt {3}\right ) \]
✓ Solution by Mathematica
Time used: 0.394 (sec). Leaf size: 76
DSolve[y'[x]+2*y[x]-y[x]^2==-2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-\left (\sqrt {3}-1\right ) e^{2 \sqrt {3} (x+c_1)}+1+\sqrt {3}}{1+e^{2 \sqrt {3} (x+c_1)}} \\ y(x)\to 1-\sqrt {3} \\ y(x)\to 1+\sqrt {3} \\ \end{align*}