Practice Method Of Separation Of Variables (16.11) - Partial Differential Equations – Basic Concepts

Practice: Method of Separation of Variables

Lesson 16.11Partial Differential Equations – Basic Concepts

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10 questions

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6 sample questions

Question 1.
Define the method of separation of variables in your own words.
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Consider how we might write the solution format.
Question 2.
What is the first step in applying this method?
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Focus on separating the variables.
3Question 3.
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)
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Think about how variables are combined.
4Question 4.
True or False: The method of separation of variables can only be used for second-order PDEs.
- True
- False
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Consider the flexibility of the method.
Question 5.
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.
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Follow through with the separation and ensure to use boundary conditions.
Question 6.
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.
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Consider Fourier series for $f(x)$ and $g(x)$.

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