Differential Equation Help RL Circuit Alternating

Question

I need to solve for the current, $i$, that satisfies the differential equation: $$L\frac{\mathrm{d}i}{\mathrm{d}t}+Ri=E$$ where $$E=\sin(2t)$$ I'd be able to do it if $E$ was a constant but I'm struggling with dealing with it as a function. At the moment, I'm not sure how to get both the $\mathrm{d}t$ and $\sin(2t)$ to the one side of the equation.

Answer 1

You should use Laplace transform: solve it in frequency domain and then apply Inverse Laplace formulas to get the answer. This approach will also bring you a physical point of view (frequency of the phenomena and so on)

Answer 2

Well, when we have a series RL circuit we know that:

$$\text{V}_{\space\text{in}}\left(t\right)=\text{V}_{\space\text{R}}\left(t\right)+\text{V}_{\space\text{L}}\left(t\right)\tag1$$

Now, for the resistor we can use:

$$\text{V}_{\space\text{R}}\left(t\right)=\text{I}_{\space\text{R}}\left(t\right)\cdot\text{R}\tag2$$

And for the coil:

$$\text{V}_{\space\text{L}}\left(t\right)=\text{I}_{\space\text{L}}'\left(t\right)\cdot\text{L}\tag3$$

And we know that the current in both the resistor and inductor are equal, so:

$$\text{I}_{\space\text{in}}\left(t\right)=\text{I}_{\space\text{R}}\left(t\right)=\text{I}_{\space\text{L}}\left(t\right)\tag4$$

When we have an AC circuit, we can set:

$$\text{V}_{\space\text{in}}\left(t\right)=\hat{\text{u}}\cdot\sin\left(\omega\cdot t+\varphi\right)\tag5$$

So, now we can write:

$$\hat{\text{u}}\cdot\sin\left(\omega\cdot t+\varphi\right)=\text{I}_{\space\text{in}}\left(t\right)\cdot\text{R}+\text{I}_{\space\text{in}}'\left(t\right)\cdot\text{L}\space\Longleftrightarrow$$ $$\text{I}_{\space\text{in}}\left(t\right)=\exp\left(-\frac{\text{R}}{\text{L}}\cdot t\right)\cdot\left\{\frac{1}{\text{L}}\int\exp\left(\frac{\text{R}}{\text{L}}\cdot t\right)\cdot\hat{\text{u}}\cdot\sin\left(\omega\cdot t+\varphi\right)\space\text{d}t+\text{K}\right\}\tag6$$