Pacemaker differential equation

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SUMMARY

The discussion revolves around solving a differential equation related to a pacemaker circuit consisting of a battery, capacitor, and heart resistance. The key equation to determine is E(t), which represents the electric tension applied to the heart. The participants explore whether this equation is a first-order linear and homogeneous equation. Ultimately, the solution involves separating variables and differentiating, confirming that the initial voltage can be derived from the established relationships within the circuit.

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear equations.
  • Knowledge of electrical circuits, including components like capacitors and resistors.
  • Familiarity with the concepts of voltage, current, and resistance in electrical engineering.
  • Ability to interpret circuit diagrams and analyze their components.
NEXT STEPS
  • Study the properties of first-order linear differential equations.
  • Learn about the behavior of capacitors in DC circuits.
  • Research methods for solving differential equations through variable separation.
  • Explore the application of Kirchhoff's laws in circuit analysis.
USEFUL FOR

This discussion is beneficial for students and professionals in electrical engineering, particularly those focusing on circuit analysis and differential equations. It is also useful for anyone interested in the mathematical modeling of biological systems, such as pacemakers.

itzela
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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 resistance 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?
 

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itzela said:
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 resistance in the circuit. When the commuter S connects to P the capacitor charges,

I can't make out the commuter in the diagram. It looks like it could be just a switch that shuts out the resistor, leaving only the capacitor and inductor in series. If that is the case then the capacitor will certainly not be charged. Capacitors that are both in series and in parallel with a DC voltage source cannot be charged with that source.

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


What is t_1?

Would it be a first order linear equation?
Is it homogenous?

What is the definition of a first order equation? Of a linear equation? Of a homogeneous equation?

Once you look those up: Can you try to apply those definitions to your equation to see if it satisfies them?

What would be the value of the initial voltage?

Without a clearer diagram, I can't tell.
 
Thanks Tom... i figured it out. It was actually quite simply, just a matter of separating the variables and differentiating.
 

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