How Do Differential Equations Model Electric Circuits in Medical Devices?

[itzela]

Hi Guys... I'm trying to learn diff.eq on my own and I'm stuck on a problem and I don't even know where or how to begin:

the problem is: the pacemaker shown in the figure (first attatchment) is made up of an electric battery, a small capacitor, and the heart which functions like a resistence in the circuit. When the commuter S connects to P the capacitor charges, when S is connected to Q the capacitor discharges sending an electric shock to the heart. During this time the electric tension E applied to the heart is given by: (second attatchment).

The resistance and the capacitance are both constant...
what is:
E(t)= ? E(t1) = Eo

Would it be a first order linear equation?
Is it homogenous?
What would be the value of the initial voltage?

- i translated it directly from spanish so if further clarifying of the problem is needed please let me know.

 

[itzela]

any advice on how to start?

 

[kyle8921]

Hello,

I can help you with your problem. Diff.eq, or differential equations, is a mathematical tool used to describe physical systems and phenomena. In this case, we are dealing with an electric circuit involving a pacemaker and a heart.

To begin solving this problem, we need to understand the components of the circuit. The electric battery provides a constant voltage, which we can call Eo. The small capacitor has a constant capacitance, C, and the heart acts as a resistance, R.

When the commuter S is connected to P, the capacitor charges up to the voltage of the battery, Eo. When S is then connected to Q, the capacitor discharges, sending an electric shock to the heart. This discharge causes a change in the electric tension applied to the heart, which we can denote as E(t).

Based on the information given, we can write the following equation:

E(t) = Eo * (1 - e^(-t/RC))

This is a first-order linear differential equation, as it involves the first derivative of E(t). It is also homogeneous, as the equation only contains E(t) and its derivatives.

To find the initial voltage, E(t1), we need to know the time, t1, when the commuter S is connected to Q. This will depend on the specific settings of the pacemaker and the heart's response time. Once we know t1, we can plug it into the equation above to find E(t1).

I hope this helps you with your problem. If you need further clarification or assistance, please let me know. Good luck with your studies!