Practice Case 2: Repeated Real Roots (D =0) - 3.3.2 | 3. Second-Order Homogeneous Equations with Constant Coefficients | Mathematics (Civil Engineering -1)
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3.3.2 - Case 2: Repeated Real Roots (D =0)

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Write the general solution for a differential equation with repeated roots.

💡 Hint: Use the form that includes a polynomial term.

Question 2

Easy

What does a discriminant of zero indicate?

💡 Hint: Think about the nature of roots in polynomials.

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Interactive Quizzes

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

Question 1

What form does the general solution take for repeated roots?

  • y(x) = C_1 e^{rx}
  • y(x) = (C_1 + C_2 x)e^{rx}
  • y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x}

💡 Hint: Think about the role of x in the solution.

Question 2

Is the discriminant zero when the roots are complex?

  • True
  • False

💡 Hint: Recall the conditions for different types of roots.

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Challenge Problems

Push your limits with challenges.

Question 1

Given the differential equation y'' - 6y' + 9y = 0, analyze how varying initial conditions of y(0) = 5 and y'(0) = 0 affect the constants and the resultant function.

💡 Hint: Apply the initial conditions directly for each constant.

Question 2

Discuss in detail how repeated roots could relate to real-world scenarios, for instance, in the context of earthquake-resistant structures.

💡 Hint: Consider how structures respond to forces over time with repeated root solutions.

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