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

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Write the auxiliary equation for $y'' - 3y' + 2y = 0$.

💡 Hint: Replace each derivative with powers of m.

Question 2

Easy

What is the general solution for $y'' - 4y = 0$?

💡 Hint: Consider the auxiliary equation and the roots.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

What is the general form of the auxiliary equation derived from $y'' - 3y' + 2y = 0$?

  • $m^{2} - 3m + 2 = 0$
  • $m^{2} + 3m + 2 = 0$
  • $2m^{2} - 3m + 2 = 0$

💡 Hint: Replace each derivative with m raised to its order.

Question 2

True or False: The general solution of a second-order linear homogeneous equation is a combination of two independent solutions.

  • True
  • False

💡 Hint: Consider how solutions are structured in linear differential equations.

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Challenge Problems

Push your limits with challenges.

Question 1

Consider the equation $y'' + 4y = 0$. Find the roots of the auxiliary equation and determine the general solution. Discuss the implications of the roots.

💡 Hint: Recall how to interpret complex roots in the general solution.

Question 2

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 $y'(0) = 0$.

💡 Hint: Remember to differentiate the general solution when a double root is present to apply the initial conditions.

Challenge and get performance evaluation