2.3 problem 3

Internal problem ID [3142]
Internal file name [OUTPUT/2634_Sunday_June_05_2022_08_37_47_AM_71070688/index.tex]

Book: An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section: Chapter 4, Ex. 4.2
Problem number: 3.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program :

Maple gives the following as the ode type

[_separable]

\[ \boxed {r y^{\prime }-\frac {\left (a^{2}-r^{2}\right ) \tan \left (y\right )}{a^{2}+r^{2}}=0} \]

2.3.1 Solving as separable ode

In canonical form the ODE is \begin {align*} y' &= F(r,y)\\ &= f( r) g(y)\\ &= \frac {\left (a^{2}-r^{2}\right ) \tan \left (y \right )}{\left (a^{2}+r^{2}\right ) r} \end {align*}

Where \(f(r)=\frac {a^{2}-r^{2}}{\left (a^{2}+r^{2}\right ) r}\) and \(g(y)=\tan \left (y \right )\). Integrating both sides gives \begin{align*} \frac {1}{\tan \left (y \right )} \,dy &= \frac {a^{2}-r^{2}}{\left (a^{2}+r^{2}\right ) r} \,d r \\ \int { \frac {1}{\tan \left (y \right )} \,dy} &= \int {\frac {a^{2}-r^{2}}{\left (a^{2}+r^{2}\right ) r} \,d r} \\ \ln \left (\sin \left (y \right )\right )&=-\ln \left (a^{2}+r^{2}\right )+\ln \left (r \right )+c_{1} \\ \end{align*} Raising both side to exponential gives \begin {align*} \sin \left (y \right ) &= {\mathrm e}^{-\ln \left (a^{2}+r^{2}\right )+\ln \left (r \right )+c_{1}} \end {align*}

Which simplifies to \begin {align*} \sin \left (y \right ) &= c_{2} {\mathrm e}^{-\ln \left (a^{2}+r^{2}\right )+\ln \left (r \right )} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \arcsin \left (\frac {c_{2} {\mathrm e}^{c_{1}} r}{a^{2}+r^{2}}\right ) \\ \end{align*}

Verification of solutions

\[ y = \arcsin \left (\frac {c_{2} {\mathrm e}^{c_{1}} r}{a^{2}+r^{2}}\right ) \] Verified OK.

2.3.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & r y^{\prime }-\frac {\left (a^{2}-r^{2}\right ) \tan \left (y\right )}{a^{2}+r^{2}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a^{2}-r^{2}\right ) \tan \left (y\right )}{\left (a^{2}+r^{2}\right ) r} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\tan \left (y\right )}=\frac {a^{2}-r^{2}}{\left (a^{2}+r^{2}\right ) r} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} r \\ {} & {} & \int \frac {y^{\prime }}{\tan \left (y\right )}d r =\int \frac {a^{2}-r^{2}}{\left (a^{2}+r^{2}\right ) r}d r +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\sin \left (y\right )\right )=-\ln \left (a^{2}+r^{2}\right )+\ln \left (r \right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arcsin \left (\frac {{\mathrm e}^{c_{1}} r}{a^{2}+r^{2}}\right ) \end {array} \]

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: 0.015 (sec). Leaf size: 17

dsolve(r*diff(y(r),r)= (a^2-r^2)/(a^2+r^2)*tan(y(r)),y(r), singsol=all)
 

\[ y \left (r \right ) = \arcsin \left (\frac {r c_{1}}{a^{2}+r^{2}}\right ) \]

✓ Solution by Mathematica

Time used: 23.337 (sec). Leaf size: 26

DSolve[r*y'[r]== (a^2-r^2)/(a^2+r^2)*Tan[y[r]],y[r],r,IncludeSingularSolutions -> True]
                                                                                    
                                                                                    
 

\begin{align*} y(r)\to \arcsin \left (\frac {e^{c_1} r}{a^2+r^2}\right ) \\ y(r)\to 0 \\ \end{align*}