Practice Case I: Real and Distinct Roots - 2.4 | 2. Homogeneous Linear Equations of Second Order | Mathematics (Civil Engineering -1)
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Case I: Real and Distinct Roots

2.4 - Case I: Real and Distinct Roots

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Learning

Practice Questions

Test your understanding with targeted questions

Question 1 Easy

What is the general form of a second-order linear homogeneous equation?

💡 Hint: Look for terms involving y and its derivatives.

Question 2 Easy

Define a homogeneous differential equation.

💡 Hint: Focus on the equality condition.

4 more questions available

Interactive Quizzes

Quick quizzes to reinforce your learning

Question 1

What form does the general solution take when the roots are real and distinct?

y(x) = C1 e^(m1 x) + C2 e^(m2 x)
y(x) = (C1 + C2 x)e^(m x)
y(x) = e^(αx)(C cos(βx) + C sin(βx))

💡 Hint: Recall the response of distinct roots.

Question 2

True or False: The solution to a second-order linear homogeneous equation can always be expressed using two arbitrary constants.

True
False

💡 Hint: Think about how we determine specific solutions.

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Challenge Problems

Push your limits with advanced challenges

Challenge 1 Hard

Given the equation d²y/dx² + 4dy/dx + 5y = 0, determine if the roots are real and distinct. Solve for y(x).

💡 Hint: Calculate the discriminant.

Challenge 2 Hard

If the auxiliary equation yields roots m1 = 3 and m2 = 5, find the specific solution if given initial conditions y(0) = 1, y'(0) = 0.

💡 Hint: Express C1 in terms of C2 using the first initial condition.

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