Internal problem ID [12974]
Internal file name [OUTPUT/11627_Tuesday_November_07_2023_11_52_33_PM_86942080/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 1.6 page 89
Problem number: 12.
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]
\[ \boxed {w^{\prime }-\left (w^{2}-2\right ) \arctan \left (w\right )=0} \]
Integrating both sides gives \begin {align*} \int _{}^{w}\frac {1}{\left (\textit {\_a}^{2}-2\right ) \arctan \left (\textit {\_a} \right )}d \textit {\_a} = t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{w}\frac {1}{\left (\textit {\_a}^{2}-2\right ) \arctan \left (\textit {\_a} \right )}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{w}\frac {1}{\left (\textit {\_a}^{2}-2\right ) \arctan \left (\textit {\_a} \right )}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & w^{\prime }-\left (w^{2}-2\right ) \arctan \left (w\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & w^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & w^{\prime }=\left (w^{2}-2\right ) \arctan \left (w\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {w^{\prime }}{\left (w^{2}-2\right ) \arctan \left (w\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {w^{\prime }}{\left (w^{2}-2\right ) \arctan \left (w\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {w^{\prime }}{\left (w^{2}-2\right ) \arctan \left (w\right )}d t =t +c_{1} \end {array} \]
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: 0.0 (sec). Leaf size: 25
dsolve(diff(w(t),t)=(w(t)^2-2)*arctan( w(t) ),w(t), singsol=all)
\[ t -\left (\int _{}^{w \left (t \right )}\frac {1}{\left (\textit {\_a}^{2}-2\right ) \arctan \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.909 (sec). Leaf size: 62
DSolve[w'[t]==(w[t]^2-2)*Arctan[ w[t]],w[t],t,IncludeSingularSolutions -> True]
\begin{align*} w(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {Arctan}(K[1]) \left (K[1]^2-2\right )}dK[1]\&\right ][t+c_1] \\ w(t)\to -\sqrt {2} \\ w(t)\to \sqrt {2} \\ w(t)\to \text {Arctan}^{(-1)}(0) \\ \end{align*}