Practice Second-order Homogeneous Equations With Constant Coefficients (3)

Practice: Second-Order Homogeneous Equations with Constant Coefficients

Lesson 3Second-Order Homogeneous Equations with Constant Coefficients

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9 questions

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Exercise
6 questions · about 4 min · pass mark 70%. Hints available.
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3 questions · about 2 min · pass mark 70%. Hints available.
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6 sample questions

Question 1.
What is the general form of a second-order homogeneous linear differential equation?
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Look for the terms and their coefficients.
Question 2.
Define the term 'homogeneous'.
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Think about the implications of zero.
3Question 3.
What is the characteristic equation derived from the second-order linear differential equation?
- ar^2 + br + c = 0
- d^2y/dx^2 = 0
- y = C e^{kx}
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Think about substitution.
4Question 4.
True or False: The general solution of a second-order homogeneous equation can be formed with exponential terms only.
- True
- False
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Recall the nature of the roots.
Question 5.
Derive the characteristic equation for the second-order linear ODE $y'' - 7y' + 10y = 0$ , classify the roots, and find the general solution.
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Start by substituting your assumed solution.
Question 6.
Model the free vibration of a cantilever beam described by the ODE $m \frac{d^2y}{dt^2} + c \frac{dy}{dt} + k y = 0$ . Derive and solve for the response.
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Identify damping cases based on the roots.

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