Internal problem ID [13041]
Internal file name [OUTPUT/11694_Wednesday_November_08_2023_03_28_52_AM_18688578/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. Review Exercises for chapter 1. page
136
Problem number: 27.
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]
\[ \boxed {y^{\prime }-y^{2}=3} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}+3}d y &= t +c_{1}\\ \frac {\sqrt {3}\, \arctan \left (\frac {y \sqrt {3}}{3}\right )}{3}&=t +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=\sqrt {3}\, \tan \left (\left (t +c_{1} \right ) \sqrt {3}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \sqrt {3}\, \tan \left (\left (t +c_{1} \right ) \sqrt {3}\right ) \\ \end{align*}
Verification of solutions
\[ y = \sqrt {3}\, \tan \left (\left (t +c_{1} \right ) \sqrt {3}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=3 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=3+y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{3+y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{3+y^{2}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\sqrt {3}\, \arctan \left (\frac {y \sqrt {3}}{3}\right )}{3}=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\sqrt {3}\, \tan \left (\left (t +c_{1} \right ) \sqrt {3}\right ) \end {array} \]
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: 16
dsolve(diff(y(t),t)= 3+y(t)^2,y(t), singsol=all)
\[ y \left (t \right ) = \sqrt {3}\, \tan \left (\left (t +c_{1} \right ) \sqrt {3}\right ) \]
✓ Solution by Mathematica
Time used: 0.256 (sec). Leaf size: 48
DSolve[y'[t]==3+y[t]^2,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \sqrt {3} \tan \left (\sqrt {3} (t+c_1)\right ) \\ y(t)\to -i \sqrt {3} \\ y(t)\to i \sqrt {3} \\ \end{align*}