Solve the simultaneous differential equations
L(di1/dt)+Ri1+(1/2)L(di2/dt)=E
L(di2/dt)+Ri2+(1/2)L(di1/dt)=0
where 𝐿, 𝑅, and 𝐸 are constants and given that i1=i2=0 when t=o
To solve the given simultaneous differential equations, we can use the method of Laplace transforms. Here's how you can proceed:
Step 1: Take the Laplace transform of both sides of each equation. Applying the Laplace transform to each term using the linearity property, we get:
L(s * i1) + R * I1(s) + (1/2) * L(s * i2) = E * (1/s)
L(s * i2) + R * I2(s) + (1/2) * L(s * i1) = 0
Step 2: Apply the initial conditions. Since i1 = i2 = 0 when t = 0, we can represent this condition in terms of Laplace transforms as:
I1(0) = 0
I2(0) = 0
Step 3: Solve the resulting algebraic equations for I1(s) and I2(s). Rearrange the equations obtained in Step 1 to solve for I1(s) and I2(s):
(L * s + R) * I1(s) + (1/2) * L * s * I2(s) = E * (1/s)
(1/2) * L * s * I1(s) + (L * s + R) * I2(s) = 0
Step 4: Solve the equations simultaneously for I1(s) and I2(s). By rearranging and eliminating variables, we get:
(2 * (L * s + R)) * I1(s) + L * s * I2(s) = 2 * E
L * s * I1(s) + (2 * (L * s + R)) * I2(s) = 0
Step 5: Solve the resulting linear system of equations for I1(s) and I2(s). You can use various methods to solve this system, such as substitution, elimination, or matrix methods. Once the values of I1(s) and I2(s) are obtained, proceed to the next step.
Step 6: Take the inverse Laplace transform of I1(s) and I2(s) to obtain the solutions i1(t) and i2(t) in the time domain. This step requires the use of inverse Laplace transform techniques, such as partial fractions, tables, or software.
By following these steps, you can solve the given simultaneous differential equations and find the solutions i1(t) and i2(t).