Anonymous asked in Science & MathematicsPhysics · 1 decade ago

Ldi/dt + ir -Vcos wt=0 what technique do i use to solve?

RL AC circuit

2 Answers

  • JSAM
    Lv 5
    1 decade ago
    Favorite Answer

    You need to use an integrating factor to solve this differential equation. First re-write the equation as:

    di/dt + i(R/L) = (V/L)*cos(wt)

    On the left hand side, you see that we have the original variable 'i' and the derivative. To solve for i(t), we need to re-write the LHS into a derivative form (di/dt) and integrate. We need to define an integrating factor to do this. Let's define the integrating factor as:

    exp(int[R/L dt]) where 'int' denotes integral. Also, let's denote it as 'u' to make simplification easier.

    Next, multiply the above equation by 'u':

    u(di/dt) + i*u(R/L) = u*(V/L)*cos(wt)

    The 2nd term of the left hand side can be re-written as i*(du/dt). You can check this by checking the derivative manually:

    u = exp(int[R/L dt])

    du = exp(int[R/L dt])*(R/L)--->need to use Chain rule--->= u*(R/L)

    Thus, we have:

    u(di/dt) + i(du/dt) = u*(V/L)*cos(wt)

    By inspection, you see that the LHS represents the product rule of the two functions 'u' and 'i'.

    d(i*u)/dt = u*(V/L)*cos(wt)

    Now you can integrate and solve for i(t):

    i(t) = int[u*(V/L)*cos(wt)]/u where u = exp(int[R/L dt])

    You could try to solve for the integral manually, but that would be rather difficult


    Hope this helps

  • 1 decade ago

    di/dt + (r/L)i = (V/L)cos wt

    When in this form we can find the integrating factor u(x) where

    u(x) = exp [integral (r/L) dt]

    Now note how by multiplying throughout by u(x) the LHS can be made exact and an application of integration on the RHS gives the general solution.

    Hope this helps!

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