Solve the following differential equations subject to the specified initial conditions
- (a) d2v/dt2 + 4v = 12, v(0) = 0, dv(0)/dt = 2
- (b) d2i/dt2 + 5 di/dt + 4i = 8, i(0) = −1, di(0)/dt = 0
- (c) d2v/dt2 + 2 dv/dt + v = 3, v(0) = 5, dv(0)/dt = 1
- (d) d2i/dt2 + 2 di/dt + 5i = 10, i(0) = 4, di(0)/dt = −2
(a)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the voltage response with the step input, if the roots of characteristic equation are real and imaginary.
Here,
Write a general expression for the roots of characteristic equation when the roots are real and imaginary.
Write an expression to solve quadratic equation.
Here,
Calculation:
From equation (1), the characteristic equation is written as follows.
From the equation (6),
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Compare the values of roots with the equation (3) and equation (4).
Substitute
At
Therefore,
Substitute
For the step response
Substitute
Substitute
Substitute
Substitute
Substitute
Differentiate equation (11) with respect to
Substitute
Substitute
Substitute
Conclusion:
Thus, the expression of
(b)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write a general expression for the current response with the step input, if the roots of characteristic equation are real and negative.
Here,
Calculation:
From equation (14), the characteristic equation is written as follows.
Compare the equation (16) with the quadratic equation
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and negative.
Substitute
At
Therefore,
Substitute
For the step response
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Differentiate equation (19) with respect to
Substitute
Substitute
Substitute equation (20) in above equation to find
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the expression of
(c)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the voltage response with the step input, if the roots of characteristic equation are real and equal.
Here,
Write a general expression for the roots of characteristic equation when the roots are real and equal.
Calculation:
From equation (21), the characteristic equation is written as follows.
Compare the equation (24) with the quadratic equation
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Substitute
Substitute
Substitute
For the step response,
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Expand the equation (27) as follows:
Differentiate the above with respect to
Substitute
Substitute
Simplify the above equation to find
Substitute
Substitute
Conclusion:
Thus, the expression of
(d)
Find the expression of
Answer to Problem 29P
The expression of
Explanation of Solution
Given data:
The differential equation is,
The values of initial conditions are,
Formula used:
Write an expression to find the current response with the step input, if the roots of characteristic equation are real and imaginary.
Calculation:
From equation (28), the characteristic equation is written as follows.
From the equation (6),
Substitute
Simplify the equation.
Therefore, the roots of characteristic equations are real and imaginary.
Compare the values of roots with the equation (3) and equation (4).
Substitute
Substitute
For the step response,
Substitute
Substitute
Substitute
Substitute
Simplify the above equation to find
Substitute
Differentiate equation (34) with respect to
Substitute
Substitute
Substitute
Conclusion:
Thus, the expression of
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Chapter 8 Solutions
Fundamentals of Electric Circuits
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