
(a)
The solution of one loop circuit with
(a)

Answer to Problem 1RE
The solution of one-loop L-R-C circuit is,
Explanation of Solution
Result used:
From Kirchhoff’s voltage law,
From the above equation, the system of differential equations
The initial condition
Here,
Calculation:
Given that, the condition
The initial condition is
Substitute the respective values in the system of equation
Simplify the system of equations as follows.
From above system of equation, the solution in matrix form as follows.
Consider the vector
Let
Simplify further as,
Thus, the corresponding eigenvectors are
Use the eigenvalues and eigenvectors and compute fundamental matrix as,
Compute the inverse matrix of
Now compute inverse matrix as follows.
Thus, the value of
Note that,
Simplify further as,
The solution of one-loop L-R-C circuit is,
b.
The solution of one loop circuit with
b.

Answer to Problem 1RE
The solution of one-loop L-R-C circuit is,
Explanation of Solution
Given that, the condition
The initial condition is
Substitute the respective values in the system of equation
On further simplification.
From above system of equation, the solution in matrix form as follows.
Consider the vector
Let
Simplify further,
Thus, the corresponding eigenvectors are
Use the eigenvalues and eigenvectors and compute fundamental matrix as,
Compute the inverse matrix of
Now compute inverse matrix as follows.
Thus, the value of
Here, the matrix
Simplify further as,
The solution of one-loop L-R-C circuit is,
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Chapter 7 Solutions
Introductory Differential Equations
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