Solution 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.