# Ordinary Differential equation help needed.

• Nov 19th 2007, 06:58 AM
naisy18
Ordinary Differential equation help needed.
w = omega, just cant get it on.

ii) d2i/dt2 + 900i = A0 sin (wt)

Assuming that w2 not equal to 900, determine the current i in terms of the parameters (w and A0) and the variable t when the initial conditions are

i (0) = di/dt (0) = 0.

iii) Repeat part (ii) for w2 = 900

Help required people.
• Nov 19th 2007, 04:45 PM
kalagota
Quote:

Originally Posted by naisy18
w = omega, just cant get it on.

ii) d2i/dt2 + 900i = A0 sin (wt)

Assuming that w2 not equal to 900, determine the current i in terms of the parameters (w and A0) and the variable t when the initial conditions are

i (0) = di/dt (0) = 0.

iii) Repeat part (ii) for w2 = 900

Help required people.

i was able to wake up early.. Ü

$\frac{d^2i}{dt^2} + 900i = A_0 sin (\omega t)$

let us try solving the general case: $\frac{d^2y}{dt^2} + \omega ^2y = 0$

consider the operator $D = \frac{d}{dt}$, then we can transform the equation into $D^2 y + \omega ^2 y = (D^2 + \omega ^2)y = (D - \omega i)(D + \omega i)y = 0$, where $i^2 = -1$ (i have to change our variable to y because we need the imaginary number $i$).

so, $(D - \omega i)y = \left( {\frac{d}{dt} - \omega i} \right)y=0$

but the middle one is also equal to $e^{i\omega t} \frac{d}{dt}\left( {e^{-i\omega t}} \right) \implies y_1 = e^{i\omega t} = cos (\omega t) + i sin (\omega t)$

also using similar arguments, it can be shown that $(D + \omega i)y \implies y_2 = cos (\omega t) - i sin (\omega t)$

if we take $\Upsilon _1 = \frac{y_1 + y_2}{2} = cos (\omega t)$ and $\Upsilon _2 = \frac{y_1 - y_2}{2i} = sin (\omega t)$, then we have the general solution:

$y(t) = c_1 cos (\omega t) + c_2 sin (\omega t)$ ---> 1

now, to solve for the value of $c_1$ and $c_2$, just consider that $y(0) = 0 = \frac{dy}{dt}(0)$

equate equation 1 to 0.
solve for the first derivative of that equation 1 and equate to 0..
now you have two equations with two unknowns .. can you contnue it?
i'm late for my class...