Practice Solved Examples - 3.7 | 3. Second-Order Homogeneous Equations with Constant Coefficients | Mathematics (Civil Engineering -1)
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3.7 - Solved Examples

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Solve the equation d²y/dx² + 3dy/dx + 2y = 0 with initial conditions y(0)=1, y′(0)=0.

💡 Hint: Use the characteristic equation to find the roots.

Question 2

Easy

Identify the roots of the characteristic equation from d²y/dx² + y = 0.

💡 Hint: Use the quadratic formula to identify the nature of 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 form does the general solution take for distinct real roots?

  • y = C₁e^(r₁x) + C₂e^(r₂x)
  • y = (C₁ + C₂x)e^(r₁x)
  • y = e^(αx)(C₁cos(βx) + C₂sin(βx))

💡 Hint: Remember the scenario with distinct roots yields this format.

Question 2

True or False: The roots of the characteristic equation dictate the form of the general solution.

  • True
  • False

💡 Hint: Consider what the roots indicate about the solution types.

Solve 1 more question and get performance evaluation

Challenge Problems

Push your limits with challenges.

Question 1

Prove that the differential equation d²y/dx²−5dy/dx+6y=0 leads to a general solution embodying distinct roots when initial conditions are set to y(0)=4 and y′(0)=3.

💡 Hint: Start with the characteristic equation and follow through with initial conditions.

Question 2

For the equation d²y/dx² + 6dy/dx + 9y=0, explore how repeated roots influence the solution's composition with initial values y(0)=1 and y′(0)=0.

💡 Hint: Recognize that repeated roots change the solution structure considerably.

Challenge and get performance evaluation