I just want to get your opinion for the best way to solve the below differential equation. The variable is r and the parameters are L1, C1, Vt, k, I0. Time is t.
Ofer A
Does not look too simple for analytical solution. Maybe you can divide by cos t, then you get d2r/dt2+Cr=(1-2*L1)C2* tg t *dr/dt+f(t). By C, C2 I denoted some constants.
Consider the case 2*L1=1 first. Then you can easily find analytical solution of homogeneous equation and there is formula for heterogeneous (look at books). Then I would use perturbation theory for small deviations of L from ½. It is also important to know applications, what are small parameters. This is my first glance.
I would not apply numerical methods unless you define parameter range first. You also have to solve Cauchi problem with 2 initial conditions.
If you guess the first solution! then you can find the second one.
check this link: i was trying to solve it online:
http://www.wolframalpha.com/input/?i=y''(t)+s+cos(t)%2By'(t)+(1-2s)+sint+%2By(t)%3Dg(t)
Well for me, I prefer to use the numerical methods.
Your equation isn't stiff (2nd order linear differential equation), so it is easy to solve by using 'ode45' in matlab; if you want to get an expression for the solved equation, you can use curve fitting.
Jamal, now computer often suggests some solution that one can copy without understanding where it comes from.
Ofer, in the past electrical engineers in good universities studied the theory of functions of complex variables. See, for example: http://www.math.ku.dk/noter/filer/koman-12.pdf
Indeed, if you would present this equation in complex form, with complex exponent, exp(iwt), and not trigonometric functions, sin(wt), cos(wt), it would be much easier to guess the solution. The 1st differentiation transforms sin(wt) into cos(wt), while the 2nd returns to sin(wt). Without the 1st derivative it describes non-dissipating waves in electric circuit (made of L,C) with an external force, while 1st derivative stands for damping on resistance R. See for example http://www2.ph.ed.ac.uk/~egardi/MfP3-Dynamics/Dynamics_lecture_10.pdf
Muhammad, the problem of numerical solutions it that you cannot solve for all parameter values. Moreover, it may be interesting to find asymptotics of the solution for time going to infinity, and it may take long time to find out. And while computers are fast now, one can get lost in all discovered graphs and found no regularity. That is why it is always better to start analytically. But now students are prepared more to play with computer programs comparing to the old time (before 1990s) when mathematicians were very skilled in analytics, also because of low access to computers and low speed.
If Vt=0 then your equation becomes linear and has a general solution in terms of hypergeometric functions, see the link below.
http://www.wolframalpha.com/input/?i=r%27%27%28t%29*cos%28t%29*L1%2Br%27%28t%29*%281-2*L1%29*sin%28t%29%2Br%28t%29*%281-1%2FC1%2BL1%29*cos%28t%29%3D0
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