Practice Example (1.5.3) - Linear Differential Equations - Mathematics (Civil Engineering -1)

Practice: Example

Lesson 1.5.3Linear Differential Equations

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6 sample questions

Question 1.
Write the auxiliary equation for $y'' - 3y' + 2y = 0$ .
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Replace each derivative with powers of m.
Question 2.
What is the general solution for $y'' - 4y = 0$ ?
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Consider the auxiliary equation and the roots.
3Question 3.
What is the general form of the auxiliary equation derived from $y'' - 3y' + 2y = 0$ ?
- $m^{2} - 3m + 2 = 0$
- $m^{2} + 3m + 2 = 0$
- $2m^{2} - 3m + 2 = 0$
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Replace each derivative with m raised to its order.
4Question 4.
True or False: The general solution of a second-order linear homogeneous equation is a combination of two independent solutions.
- True
- False
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Consider how solutions are structured in linear differential equations.
Question 5.
Consider the equation $y'' + 4y = 0$ . Find the roots of the auxiliary equation and determine the general solution. Discuss the implications of the roots.
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Recall how to interpret complex roots in the general solution.
Question 6.
Solve the differential equation $y'' + 12y' + 36y = 0$ and find particular values for $C_1$ and $C_2$ given initial conditions $y(0) = 2$ and $y'(0) = 0$ .
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Remember to differentiate the general solution when a double root is present to apply the initial conditions.

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