Separation of Variables (Worksheet)

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Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

Learning Objectives

• Identify separable simple differential equations.
• Solve separable differential equations and initial value problems.
• Determine the interval(s) (with respect to the independent variable) on which a solution to a separable differential equation is defined.

The "separation of variables" is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of an equation. The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions.

Separable Differential Equations

A separable differential equation is a differential equation whose algebraic structure permits the variables present to be separated in a particular way. For instance, consider the equation

\[\dfrac{dy}{dt} = t y.\]

We would like to separate the variables \(t\) and \(y\) so that all occurrences of \(t\) appear on the right-hand side, and all occurrences of \(y\) appears on the left and multiply \(dy/dt\). We may do this in the preceding differential equation by dividing both sides by \(y\):

\[\dfrac{1}{y} \dfrac{dy}{dt} = t. \]

Note particularly that when we attempt to separate the variables in a differential equation, we require that the left-hand side be a product in which the derivative \(dy/dt\) is one term. Not every differential equation is separable. For example, if we consider the equation

\[\dfrac{dy}{ dt} = t − y,\]

it may seem natural to separate it by writing

\[y + \dfrac{dy}{dt} = t.\]

As we will see, this will not be helpful since the left-hand side is not a product of a function of \(y\) with \(\frac{dy}{dt}\).

Q1

Which of the following differential equations are separable? If the equation is separable, write the equation in the revised form \(g(y) \dfrac{dy}{dt} = h(t).\)

1. \( \dfrac{dy}{dt} = −3y\)
2. \( \dfrac{dy}{dt}t = t y − y. \)
3. \( \dfrac{dy}{dt} = t + 1. \)
4. \( \dfrac{dy}{dt} = t^2 − y^2 .\)

Why do we include the term “\(+C\)” in the expression \[\int x\, dx = \dfrac{x^2}{2} + C?\]

Suppose we know that a certain function \(f\) satisfies the equation \[\int f'(x) \,dx = \int x\, dx.\] What can you conclude about \(f\)?

Q2

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 \(x\)?

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

What is \( \dfrac{\partial f}{\partial x}\)?

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

What is \(b(t)\)?

Q2

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 \(u(x, t ) = X(x)T (t )\).

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

Q3

The equation for the vibration of a string is (\(x\) is the distance along the string and \(t\) is time)

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

Given that \(u(x, t ) = X(x)T(t)\), the following questions will guide you through how to 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 \(t\):

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 \(t\):

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?

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