# How to find the differential equation of the current i₁?

I am trying to find the differential equation of this circuit:

I tried to use KVL at the left mesh which gave me

$$\E=R\cdot i+V_c\$$

From $$\V_c\$$ I can calculate $$\i_1\$$, but then I didn't know how to continue. I tried to use KVL at the right mesh, and I also tried the KVL $$\i=i_1+i_2\$$ but none of them worked.

How do I solve this circuit?

The steps that I have done:

$$E=R\cdot i+V_c~\mathrm{(1)}$$

We know that $$\i_1=C\frac{dv}{dt}\$$ so we can say that:

$$V_c=\frac{1}{C}\int{i_1}$$

Then we replace the $$\V_c\$$ in (1).

From KCL we find that $$\i=i_1+i_2\$$, so we turn $$\R\cdot i\$$ into $$\R\cdot(i_1+i_2)\$$.

Now I don't know how I can find the relation between $$\i_2\$$ and $$\i_1\$$.

As this is a "homework related" question, i will not give a full answer.

But i will give you a guide - also you are free the ask if anything is unclear.

1. From a look on your circuit one becomes obvious: A equation for the total current is helfpull. $$i(t) = i1(t) + i2(t)$$

2. The total current will split into i1 and i2 in respect to the specific impedances. So the question becomes: What are your impedances over time? $$Z1(t) , Z2(t)$$

3. If you are also interested in the steady-states: What are these ?

$$I1(t=0+), I1(t= \infty), I1(t=0-)$$

1. If you want to draw the current over time: What are your time constants? $$\tau_1,\tau_2$$

How i would do it: I would try to avoid differential equations as long as possible - they are messy in my opinion. I would try to get an equation for the current in question based on impedances - as soon as this equation is available to me, i would start to substitute impedances with differential equations: Many things will cancel out.

• I am student from algeria and this is not a homework it just an exercise that i have found. Our teacher told us that he will always give us the steady states in the problem and that we have only to find the constants of the solution of the differential equation. But he told told us to learn how to write differential equations so that is whh i posted this question.so can you please give me the full answer, and jf there is a way to avoid the differential equation i would like to have an idea about it Jan 22, 2023 at 19:02
• @BadrEddine Just get and equation based on impedances for your current in question. When you have your equation substitute z_c = u_c / ( du_c/dt ). Jan 22, 2023 at 19:11
• @BadrEddine Also: If you really want to shine and learn something: youtube.com/… ! This is how to avoid differential equations at all cost - really handy and quite simple actually. Jan 22, 2023 at 19:15
• Thank you very much, what about writing the differential equation? Have you got any tricks that can help me ? Jan 22, 2023 at 19:57