Partial Derivatives and Differential Equations

Section 8.1 Partial Derivatives and Differential Equations

Definition 8.1.1.

A partial derivative of \(f(x,y,z)\) with respect to \(x\) is the derivative of \(f\) with respect to \(x\) keeping the other variables constant. Technically

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\begin{equation*} \frac{\partial f(x,y,z)}{\partial x} = \lim_{\Delta x \rightarrow 0} \frac{f(x+\Delta x,y,z)-f(x,y,z)}{\Delta x} \end{equation*}

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Example 8.1.2.

If \(f(x,y) = x^2 + xy + y \sin x\text{,}\) Find \(\displaystyle \frac{\partial f}{\partial x}\) and \(\displaystyle \frac{\partial f}{\partial y}\text{.}\)

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Solution.

To find \(\frac{\partial f}{\partial x}\) consider \(y\) a constant, so

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\begin{equation*} \frac{\partial f}{\partial x} = 2x + y + y \cos x \end{equation*}

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similarly to find \(\frac{\partial f}{\partial y}\) consider \(x\) a constant,

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\begin{equation*} \frac{\partial f}{\partial y} = x + \sin x \end{equation*}

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Subsection 8.1.1 Higher-Order Partial Derivatives

As with ordinary derivatives, we can have higher-order partial derivatives. That is we define the 2nd order partial derivative of \(f\) with respect to \(x\) as

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\begin{equation*} \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \frac{\partial f}{\partial x} \end{equation*}

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that is it is the partial derivative of the partial derivative. Since ordinary derivatives involve only functions of one variable, the mixed derivative is a new concept. If \(f\) is a function of \(x\) and \(y\) or \(f(x,y)\text{,}\) then we can have the partial derivative of \(\frac{\partial f}{\partial x}\) with respect to \(y\) or \(\frac{\partial f}{\partial y}\) with respect to \(x\text{.}\) We write these as

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\begin{align*} \frac{\partial^2 f}{\partial y \partial x} \amp = \frac{\partial}{\partial y} \frac{\partial f}{\partial x} \\ \frac{\partial^2 f}{\partial x \partial y} \amp = \frac{\partial}{\partial x} \frac{\partial f}{\partial y} \end{align*}

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Example 8.1.3.

Find all second-order derivatives of

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\begin{equation*} f(x,y) = x^3 + x^2y + xe^y \end{equation*}

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Solution.

First, let’s find the two first derivatives:

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\begin{align*} \frac{\partial f}{\partial x} \amp = 3x^2 + 2xy + e^y \\ \frac{\partial f}{\partial y} \amp = x^2 + xe^y \end{align*}

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And then there are four 2nd-order derivatives:

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\begin{align*} \frac{\partial^2 f}{\partial x^2} \amp = \frac{\partial}{\partial x} \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (3x^2 + 2xy + e^y) = 6x + 2y \\ \frac{\partial^2 f}{\partial y \partial x} \amp = \frac{\partial}{\partial y} \frac{\partial f}{\partial x} = \frac{\partial}{\partial y} (3x^2+2xy+e^y) = 2x + e^y \\ \frac{\partial^2 f}{\partial x \partial y} \amp = \frac{\partial}{\partial x} \frac{\partial f}{\partial y} = \frac{\partial}{\partial x} (x^2+xe^y) = 2x + e^y \\ \frac{\partial^2 f}{\partial y^2} \amp = \frac{\partial}{\partial y} \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (x^2+xe^y) = xe^y \end{align*}

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You probably noticed that the two derivatives involving both \(x\) and \(y\) resulted in the same results. This is true for most functions as is shown in the next theorem:

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Theorem 8.1.4. Clairaut’s Theorem.

If \(f(x_1, x_2, \ldots, x_n)\) is a real valued function with continuous second derivatives at the point \((a_1, a_2, \ldots, a_n)\text{,}\) then

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\begin{equation*} \frac{\partial^2 f}{\partial x_i \partial x_j}(a_1,a_2, \ldots, a_n) = \frac{\partial^2 f}{\partial x_j \partial x_i} (a_1,a_2,\ldots, a_n) \end{equation*}

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In other words, partial derivatives commute.

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Subsection 8.1.2 Notation for Partial Derivatives

In terms of notation, it’s often common to use a subscript as a derivative. For example \(f_x\) can be used instead of \(\frac{\partial f}{\partial x}\) or \(f_y\) instead of \(\frac{\partial f}{\partial y}\text{.}\) This can also be extended to higher order derivatives as the following shows:

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\begin{align*} f_{xx} \amp = \frac{\partial^2 f}{\partial x^2} \amp f_{yy} \amp = \frac{\partial^2 f}{\partial y^2} \\ f_{yx} \amp = \frac{\partial^2 f}{\partial x \partial y} \amp f_{xy} \amp = \frac{\partial^2 f}{\partial y \partial x} \\ f_{xxx} \amp = \frac{\partial^3 f}{\partial x^3} \amp f_{yyx} \amp = \frac{\partial^2 f}{\partial x\partial y^2} \end{align*}

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and note that the order on the variables switch between the two notations. This generally isn’t a problem because of Clairaut’s Theorem says that the derivative is independent of the order taken on the derivatives.

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Subsection 8.1.3 Differential Equations

Definition 8.1.5.

The following are related definitions.

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A differential equation is an equation that contains derivatives.
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If the differential equation has partial derivatives, then the equation is called a partial differential equation.
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If the differential equation has standard or ordinary derivatives, then the equation is called a ordinary differential equation.
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The order of the differential equation is the highest degree of any derivative.
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If the equation is linear in \(u\text{,}\) the dependent variable and its derivatives, then the equation is linear, if not, it is nonlinear.
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Lastly, a linear differential equation is called homogeneous if the only nonzero terms in the equation contain the dependent variable. If a differential equation is not homogeneous, then it is called nonhomogeneous.
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Example 8.1.6.

The following are ordinary differential equations.

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\(\displaystyle \displaystyle y' = x\)
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\(\displaystyle y' = \sin x\text{.}\)
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\(\displaystyle \displaystyle \frac{dy}{dx} = y^2\)
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\(\displaystyle y'' + 3y' + y = 0 \text{.}\)
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Equations 1,2, and 4 are linear and 3 is nonlinear. Note that even though \(\sin x\) is not a linear function, in order for a differential equation to be linear, it needs only be linear in its dependent variables (in all of these examples, \(y\) is dependent). Equations 1 and 2 are nonhomogeneous and equation 4 is homogeneous. Also, the first three are first-order equations and the 4th equation is 2nd order. The following are partial differential equations:

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\(\displaystyle \displaystyle \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 0\)
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\(\displaystyle u_{t} -u u_x = 0\text{.}\)
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\(\displaystyle \displaystyle \frac{\partial u}{\partial x} + \biggl(\frac{\partial^2 u}{\partial {t}^2} \biggr)^2 = 0 \)
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\(\displaystyle u_{t} +u_y - u_{xxx} = 0\text{.}\)
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And the first and fourth equations are linear with the other two being nonlinear. The first and second equations above are first order, the 3rd equation is 2nd order and the fourth is a third-order PDE.

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