Practice Characteristic Equation (3.2) - Second-Order Homogeneous Equations with Constant Coefficients

Practice: Characteristic Equation

Lesson 3.2Second-Order Homogeneous Equations with Constant Coefficients

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8 questions

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

Question 1.
What is the characteristic equation derived from the equation d²y/dx² - 3dy/dx + 2y = 0?
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Only consider the coefficients of the original equation.
Question 2.
What happens when the discriminant is negative?
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Think about how the roots can affect the solution's behavior.
3Question 3.
What is the characteristic equation derived from d²y/dx² + 2dy/dx + y = 0?
- r² + 2r + 1 = 0
- r² - 2r + 1 = 0
- r² + 2r - 1 = 0
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Look carefully at how the differential equation coefficients correspond to terms in the characteristic equation.
4Question 4.
If the discriminant is zero, what type of roots do we have?
- True
- False
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Think about what this implies for the general solution.
Question 5.
Derive the general solution for the ODE d²y/dx² - 6dy/dx + 9y = 0.
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Use the quadratic formula and remember the implications of multiple roots.
Question 6.
For the differential equation y'' + 4y' + 20y = 0, find and analyze the nature of its roots and the general solution.
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Identify the real and imaginary parts carefully to construct the final solution.

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