Practice Method of Separation of Variables - 16.11 | 16. Partial Differential Equations – Basic Concepts | Mathematics (Civil Engineering -1)
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16.11 - Method of Separation of Variables

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

Define the method of separation of variables in your own words.

💡 Hint: Consider how we might write the solution format.

Question 2

Easy

What is the first step in applying this method?

💡 Hint: Focus on separating the variables.

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 initial assumption in the method of separation of variables?

  • a. u(x,t) = X(x) + T(t)
  • b. u(x,t) = X(x)T(t)
  • c. u(x,t) = X(x)T'(t)

💡 Hint: Think about how variables are combined.

Question 2

True or False: The method of separation of variables can only be used for second-order PDEs.

  • True
  • False

💡 Hint: Consider the flexibility of the method.

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Challenge Problems

Push your limits with challenges.

Question 1

Consider the PDE $\frac{\partial^2 u}{\partial t^2} = \alpha \frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0,t) = 0$ and $u(L,t) = 0$. Use the method of separation of variables to determine the general solution.

💡 Hint: Follow through with the separation and ensure to use boundary conditions.

Question 2

Using the wave equation, solve for $u(x,t)$ given $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$. Assume boundary conditions at both ends are zero.

💡 Hint: Consider Fourier series for $f(x)$ and $g(x)$.

Challenge and get performance evaluation