Question

An electric circuit contains a 1-H inductor, a 4-9 resistor, and a voltage source of cost. The resulting differential equation relating the current i and the time t is di/dt + 4i= cos t. Find i after 0.5 s by Euler's method with At = 0.1 s if the initial current is zero. Solve the equation exactly and compare the values. Use Euler's method to find i after 0.5 s. approx=A (Round the final answer to four decimal places as needed. Round all intermediate values to nine decimal places as needed.)

Answer 1

The given differential equation is:

di/dt + 4i = cos(t)

You can use Euler's method to find the value of i after 0.5 seconds with At = 0.1 seconds. The formula for Euler's method is:

i[n+1] = i[n] + At * [di/dt]

where i[n] is the current at the nth time step. First find di/dt:

di/dt = cos(t) - 4i

Now youcan use Euler's method with At = 0.1 seconds and i[0] = 0 to find i[5]:

i[1] = i[0] + 0.1 * [cos(0) - 4(0)] = 0.1

i[2] = i[1] + 0.1 * [cos(0.1) - 4(0.1)] = 0.154

i[3] = i[2] + 0.1 * [cos(0.2) - 4(0.154)] = 0.2046

i[4] = i[3] + 0.1 * [cos(0.3) - 4(0.2046)] = 0.2521

i[5] = i[4] + 0.1 * [cos(0.4) - 4(0.2521)] = 0.2974

Therefore, i after 0.5 seconds using Euler's method is approximately 0.2974.

Now, solve the differential equation exactly. The homogeneous equation is:

di/dt + 4i = 0

The solution to this equation is:

i(t) = Ae^(-4t)

where A is a constant. To find the particular solution, you can assume a solution of the form:

i(t) = Bcos(t) + Csin(t)

Substituting this into the differential equation, you get:

-Bsin(t) + Ccos(t) + 4Bcos(t) + 4Csin(t) = cos(t)

Matching the coefficients of cos(t) and sin(t), you get:

4B = 0 and C - B = 1

Solving for B and C, you get:

B = 0 and C = 1

Therefore, the particular solution is:

i(t) = sin(t)

The general solution is the sum of the homogeneous solution and the particular solution:

i(t) = Ae^(-4t) + sin(t)

To find A, you use the initial condition i(0) = 0:

0 = Ae^(0) + sin(0)

A = -sin(0) = 0

Therefore, the solution to the differential equation is:

i(t) = sin(t)

Using this, you can find i after 0.5 seconds:

i(0.5) = sin(0.5) ≈ 0.4794

Comparing this with the value obtained using Euler's method, you can see that the exact solution is larger.

Hope I helped you...