A heart pacemaker consists of a switch, a battery of constant voltage E0 , a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart.During the time the heart is being stimulated, the voltage E across the heart satisfies the linear differential equation
Solve the DE, subject to E(4) =E 0 .
I’ll start attacking this problem by observing that the given differential equation is separable. So i’ll do the separation as follows:
I’ll then integrate both sides of the equation as follows:
On integration we get:
Here , k is a constant of integration. I’ll then make E(t) the subject of the formula by introducing an exponential on both sides, so that:
The exponential on the right side of the equation can be broken down into two terms using the laws of indices as follows:
But ek is just a constant. Therefore ,we can rewrite the above equation as follows:
The expression for E(t) then becomes;
We are given the following boundary condition: E(4) =E 0. Plugging this into E(t),
Solving for A we get:
Therefore E(t) becomes:
Merging the two exponential terms ,we get the following:
That is the solution for E(t) in terms of the given constants and time t. The negative sign on the exponential term indicates a ‘decay’ in the voltage when the capacitor is discharging.