The electromotive force is , where is capacitance in farads, is resistance in ohms, is charge in coulombs.

Substitute in .

Substitute and .

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The differential equation is .

(a)

Graph the Directional field for the differential equation .

(b) Find the limiting value of the charge.

Find the limiting value of the charge by equating .

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The limiting value of the charge is .

(c) Is there an equilibrium solution.

Equilibrium state is the condition when becomes constant.

If is a constant then .

Substitute in .

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The equilibrium solution is .

Equilibrium solution exists and is .

(d)

The initial charge is .

Graph the directional field of

Now draw a solution curve so that it move parallel to the near by segments.

The resulting curve is solution curve which passes through .

(e) If the initial charge is , use Eulers method with step size to estimate the charge after half a second.

The differential equation is and initial condition is .

Step size .

Eulers  method:

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and .

The function is .

Substitute  and in .

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Substitute , and in .

The charge after a half second  is .

(a) Graph of the directional field for the differential equation .

(b) The limiting value of the charge is .

(c) Equilibrium solution exists and is .

(d) Graph of the directional field of :

(e) The charge after a half second  is .