Solving Differential Equation with Frequency Response

[Master1022]

Homework Statement

The AC response of an inductor can be modeled by the following differential equation:
$$L \frac{di}{dt} + iR = V$$

Find, using frequency response, the current of the system when the applied voltage V is: $$V = V_0 \sin(\omega t)$$

Homework Equations

The Attempt at a Solution

In the attached photo. Does this seem to be correct?

The final form comes out as:
$$I(t) = \frac{V_0}{R^2 + L^2\omega^2} [R\sin(\omega t) - \omega L \cos(\omega t)]$$

 

[Ray Vickson]

Homework Statement

The AC response of an inductor can be modeled by the following differential equation:
$$L \frac{di}{dt} + iR = V$$

Find, using frequency response, the current of the system when the applied voltage V is: $$V = V_0 sin(\omega t)$$

Homework Equations

The Attempt at a Solution

In the attached photo. Does this seem to be correct?

Your posted image is unreadable on my devices, so there is no way I could judge correctness. Please type out at least the final form of your solution. Doing this should be relatively straightforward using LaTex. But please remember to NOT type "sin" in LaTeX: type "\sin" instead to get much more readable results ($\sin \theta$ instead of $sin \theta,$ etc) . The same holds for most other functions: all the trig and inverse trig fcns, the logarithms "\ln" and "\log", as well as "lim", "max", "min", "sup", "inf", plus the hyperbolic fcns, etc.

 

[Master1022]

Your posted image is unreadable on my devices, so there is no way I could judge correctness. Please type out at least the final form of your solution.

Thanks for the tips. Have put the final form there now.

 

[Ray Vickson]

Thanks for the tips. Have put the final form there now.

Yes, your final form is correct for a particular solution, but the general solution needs the addition of a solution to the homogeneous DE (which dies away to zero exponentially for large $t>0$.)

Checking for correctness of a solution can always be done by plugging it into the DE to see if it works. That is something that should always be done; that used to be a tedious procedure, but nowadays we can use a computer algebra system to do all the work.