Worksheet 2 Solutions

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Example 1

Answer the following question for this function of $$t$$ and $$x$$:

$f(x, t) = e^{−3t} \cos(2x)$

• Which part(s) of $$f (x,t)$$ vary with $$x$$?

$\cos(2x)$

• Which part(s) of $$f(x,t)$$ are constant when $$x$$ is varied?

$e^{−3t}$

• What is $$\dfrac{\partial f}{\partial x}$$?

$-2e^{−3t} \sin(2x)$

• If $$f (x,t) = a(x)b(t)$$, using the definition of $$f(x,t)$$ above, what is $$a(x)$$?

$\cos(2x)$

• What is $$b(t)$$?

$e^{−3t}$

Example 2

• For any function defined as $$u(x, t) = X(x)T (t )$$, write the general expression for $$\dfrac{\partial u}{ \partial x}$$ in terms of $$u(x, t ) = X(x)T (t )$$.

$\dfrac{\partial u}{ \partial x}=T(t)\dfrac{\partial X}{ \partial x}$

• Similarly, how would you write $$\dfrac{\partial u}{\partial t}$$ in terms of $$u(x, t ) = X(x)T(t)$$?

$\dfrac{\partial u}{ \partial t}=X(x)\dfrac{\partial T}{ \partial t}$

Example 3

• The equation for the vibration of a string is ($$x$$ is the distance along the string, $$t$$ is time)

$\dfrac{\partial ^2u(x,t)}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial ^2u(x,t)}{\partial t^2}$

• Given that $$u(x, t ) = X(x)T(t)$$, how could you rewrite this equation in terms of only $$X(x)$$ and $$T(t )$$? Rewrite the equation for the vibration of a string so that one side depends only on $$x$$ and the other depends only on $$t$$:

$\dfrac{1}{X(x)}\dfrac{\partial ^2X(x)}{\partial x^2}=\dfrac{1}{v^2T(t)}\dfrac{\partial ^2T(t)}{\partial t^2}$

• Since $$x$$ and $$t$$ vary independently, each side of the equation given above must be equal to a constant, the same constant. Set both sides equal to $$K$$ (we call this the separation constant) and rewrite it as two equations, one with only $$x$$ and one with only $$t$$:

$\dfrac{\partial ^2X(x)}{\partial x^2}-KX(x)=0$

$\dfrac{\partial ^2T(t)}{\partial t^2}-Kv^2T(t)=0$

• By separating variables, we’ve managed to turn a partial differential equation in two variables into two differential equations, each with in one variable. This should be much easier to solve. What about $$u(x,t)$$ makes separation of variables possible?

$$u(x,t)$$ is a product of two functions. Each of the functions depends on one argument (x or t) only.

• Using the answer to the above question, is $$u(x,t) = x e^{-3t} \cos (2x)$$ separable? What about $$u(x,t) = e^{-ix} \cos (\dfrac{2x}{\pi t})$$? Why or why not?

$$u(x,t) = x e^{-3t} \cos (2x)$$ is separable since it is a product of two functions $$x \cos (2x)$$ and $$e^{-3t}$$ . Each of them depends on one argument only (x and t correspondingly).

$$u(x,t) = e^{-ix} \cos (\dfrac{2x}{\pi t})$$ is not separable since $$u(x,t)$$ contains the function of two arguments x and t : $$\cos (\dfrac{2x}{\pi t})$$. $$u(x,t)$$ cannot be represented as a product of the functions of single argument.

Worksheet 2 Solutions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.