Internal problem ID [12981]
Internal file name [OUTPUT/11634_Tuesday_November_07_2023_11_53_45_PM_77175554/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: 37 (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]
\[ \boxed {y^{\prime }-y \cos \left (\frac {\pi y}{2}\right )=0} \]
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {1}{\textit {\_a} \cos \left (\frac {\pi \textit {\_a}}{2}\right )}d \textit {\_a} = t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\textit {\_a} \cos \left (\frac {\pi \textit {\_a}}{2}\right )}d \textit {\_a} &= t +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {1}{\textit {\_a} \cos \left (\frac {\pi \textit {\_a}}{2}\right )}d \textit {\_a} = t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y \cos \left (\frac {\pi y}{2}\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y \cos \left (\frac {\pi y}{2}\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y \cos \left (\frac {\pi y}{2}\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {y^{\prime }}{y \cos \left (\frac {\pi y}{2}\right )}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{y \cos \left (\frac {\pi y}{2}\right )}d t =t +c_{1} \end {array} \]
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: 0.016 (sec). Leaf size: 22
dsolve(diff(y(t),t)=y(t)*cos(Pi/2*y(t)),y(t), singsol=all)
\[ t -\left (\int _{}^{y \left (t \right )}\frac {\sec \left (\frac {\pi \textit {\_a}}{2}\right )}{\textit {\_a}}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 4.801 (sec). Leaf size: 47
DSolve[y'[t]==y[t]*Cos[Pi/2*y[t]],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sec \left (\frac {1}{2} \pi K[1]\right )}{K[1]}dK[1]\&\right ][t+c_1] \\ y(t)\to -1 \\ y(t)\to 0 \\ y(t)\to 1 \\ \end{align*}