Internal problem ID [12882]
Internal file name [OUTPUT/11535_Monday_November_06_2023_01_33_06_PM_70014499/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.2. page
33
Problem number: 22.
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]
\[ \boxed {y^{\prime }-y^{2}=-4} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}-4}d y &= t +c_{1}\\ -\frac {\operatorname {arctanh}\left (\frac {y}{2}\right )}{2}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-2 \tanh \left (2 t +2 c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -2 \tanh \left (2 t +2 c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = -2 \tanh \left (2 t +2 c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=-4 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}-4 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}-4}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y^{2}-4}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y-2\right )}{4}-\frac {\ln \left (y+2\right )}{4}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {2 \left ({\mathrm e}^{4 t +4 c_{1}}+1\right )}{-1+{\mathrm e}^{4 t +4 c_{1}}} \end {array} \]
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: 0.031 (sec). Leaf size: 24
dsolve(diff(y(t),t)=y(t)^2-4,y(t), singsol=all)
\[ y \left (t \right ) = \frac {-2 c_{1} {\mathrm e}^{4 t}-2}{-1+c_{1} {\mathrm e}^{4 t}} \]
✓ Solution by Mathematica
Time used: 1.053 (sec). Leaf size: 40
DSolve[y'[t]==y[t]^2-4,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {2-2 e^{4 (t+c_1)}}{1+e^{4 (t+c_1)}} \\ y(t)\to -2 \\ y(t)\to 2 \\ \end{align*}