Practice Homogeneous Linear Equations Of Second Order (2) - Homogeneous Linear Equations of Second Order

Practice: Homogeneous Linear Equations of Second Order

Lesson 2Homogeneous Linear Equations of Second Order

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9 questions

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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.
Solve the equation $\frac{d^2y}{dx^2} + 7\frac{dy}{dx} + 12y = 0$ .
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Use the quadratic formula.
Question 2.
What is the general solution if the auxiliary equation has complex roots?
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Think about Euler's formula.
3Question 3.
What form does a second-order linear homogeneous differential equation take?
- a) \$y'' + by' + cy = 0\$
- b) \$y'' + by' = 0\$
- c) \$y' + cy = 0\$
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Look for the presence of y terms and their derivatives.
4Question 4.
True or False: The solution to all second-order linear homogeneous equations contains two arbitrary constants.
- True
- False
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Consider the degree of the equation.
Question 5.
Consider a cable suspended between two points. Assuming its deflection is described by a second-order linear homogeneous equation, derive the auxiliary equation from $y = e^{mx}$ to determine the nature of its solution.
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Think about finding the discriminant to classify roots into distinct, repeated, or complex.
Question 6.
A second-order differential equation describes the temperature distribution in a rod: $\frac{d^2T}{dx^2} - λT = 0$ . Explore solutions for different values of $λ$ and interpret their implications in real-world contexts.
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Consider how the signs of the coefficients affect the solution type.

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