Internal problem ID [536]
Internal file name [OUTPUT/536_Sunday_June_05_2022_01_43_50_AM_22554720/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 6.
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]
\[ \boxed {y^{\prime }+\frac {2 \arctan \left (y\right )}{1+y^{2}}=0} \]
Integrating both sides gives \begin {align*} \int -\frac {y^{2}+1}{2 \arctan \left (y \right )}d y &= \int {dt}\\ \int _{}^{y}-\frac {\textit {\_a}^{2}+1}{2 \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 _{}^{y}-\frac {\textit {\_a}^{2}+1}{2 \arctan \left (\textit {\_a} \right )}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {\textit {\_a}^{2}+1}{2 \arctan \left (\textit {\_a} \right )}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\frac {2 \arctan \left (y\right )}{1+y^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {2 \arctan \left (y\right )}{1+y^{2}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\left (1+y^{2}\right ) y^{\prime }}{\arctan \left (y\right )}=-2 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\left (1+y^{2}\right ) y^{\prime }}{\arctan \left (y\right )}d t =\int \left (-2\right )d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {\left (1+y^{2}\right ) y^{\prime }}{\arctan \left (y\right )}d t =-2 t +c_{1} \end {array} \]
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: 23
dsolve(diff(y(t),t) = -2*arctan(y(t))/(1+y(t)^2),y(t), singsol=all)
\[ t +\frac {\left (\int _{}^{y \left (t \right )}\frac {\textit {\_a}^{2}+1}{\arctan \left (\textit {\_a} \right )}d \textit {\_a} \right )}{2}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.013 (sec). Leaf size: 38
DSolve[y'[t] == -2*ArcTan[y[t]]/(1+y[t]^2),y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{\arctan (K[1])}dK[1]\&\right ][-2 t+c_1] \\ y(t)\to 0 \\ \end{align*}