Examples of Constructing First-Order PDEs from Families of Spheres

The PDE from family of spheres is constructed by differentiating the given family and eliminating arbitrary constants. This page gives worked examples. These examples show how the general construction becomes a practical calculation.

Suppose a family of surfaces contains arbitrary constants. To construct the corresponding first-order partial differential equation, we use the following method.

\begin{align*} \text{Given family} &\longrightarrow \text{differentiate with respect to } x \text{ and } y \nonumber \\ &\longrightarrow \text{eliminate arbitrary constants} \nonumber \\ &\longrightarrow \text{obtain a first-order PDE}. \quad \text{(1)} \end{align*}

The resulting equation must contain only

x,y,z,p,q

and fixed constants. It must not contain the arbitrary constants of the original family.

Suppose a family of surfaces contains arbitrary constants. To construct the corresponding first-order partial differential equation, we use the following method.

\begin{align*} \text{Given family} &\longrightarrow \text{differentiate with respect to } x \text{ and } y \nonumber \\ &\longrightarrow \text{eliminate arbitrary constants} \nonumber \\ &\longrightarrow \text{obtain a first-order PDE}. \quad \text{(1)} \end{align*}

The resulting equation must contain only

x,y,z,p,q

and fixed constants. It must not contain the arbitrary constants of the original family.

The construction method has three main actions: start with the given family, differentiate with respect to

x

and

y

, and eliminate the arbitrary constants. Choose a stage and click Show Step to see what that stage contributes to the final PDE. Notice that the arbitrary constants are allowed in the original family, but they must disappear from the final equation. This helps you check whether the construction has really been completed.

Show that the family of spheres

\begin{equation*} x^2+y^2+(z-c)^2=r^2 \quad \text{(2)} \end{equation*}

satisfies the first-order linear partial differential equation

\begin{equation*} y p-x q=0. \quad \text{(3)} \end{equation*}

Here

c

is the arbitrary constant, and

r

is fixed. Differentiate (2) with respect to

x

\begin{align*} 2x+2(z-c)p &=0 \nonumber \\ x+(z-c)p &=0. \quad \text{(4)} \end{align*}

Differentiate (2) with respect to

y

\begin{align*} 2y+2(z-c)q &=0 \nonumber \\ y+(z-c)q &=0. \quad \text{(5)} \end{align*}

Multiply (4) by

q

and multiply (5) by

p

. Then

\begin{align*} xq+(z-c)pq &=0, \nonumber \\ yp+(z-c)pq &=0. \quad \text{(6)} \end{align*}

Subtract the first equation in (6) from the second equation in (6). We get

\begin{align*} yp-xq &=0. \quad \text{(7)} \end{align*}

Hence the required first-order PDE is

\begin{equation*} yp-xq=0. \end{equation*}

The equation

x^2+y^2+(z-c)^2=r^2

represents a family of spheres whose centers move along the

z

-axis. Change the value of

c

, choose a fixed radius

r

, and click Update Sphere to view one member of the family. Notice that changing

c

moves the sphere up or down, but the final PDE

yp-xq=0

does not contain

c

. This visualizes why the arbitrary constant is eliminated during construction.

Show that the family of spheres

\begin{equation*} (x-a)^2+(y-b)^2+z^2=r^2 \quad \text{(8)} \end{equation*}

satisfies the first-order nonlinear partial differential equation

\begin{equation*} z^2(p^2+q^2+1)=r^2. \quad \text{(9)} \end{equation*}

Here

a

and

b

are arbitrary constants, and

r

is fixed. Differentiate (8) with respect to

x

\begin{align*} 2(x-a)+2zp &=0 \nonumber \\ x-a+zp &=0. \quad \text{(10)} \end{align*}

Differentiate (8) with respect to

y

\begin{align*} 2(y-b)+2zq &=0 \nonumber \\ y-b+zq &=0. \quad \text{(11)} \end{align*}

From (10) and (11), we get

\begin{equation*} x-a=-zp, \qquad y-b=-zq. \quad \text{(12)} \end{equation*}

Now substitute these values in (8):

\begin{align*} (x-a)^2+(y-b)^2+z^2 &=r^2 \nonumber \\ (-zp)^2+(-zq)^2+z^2 &=r^2 \nonumber \\ z^2p^2+z^2q^2+z^2 &=r^2 \nonumber \\ z^2(p^2+q^2+1) &=r^2. \quad \text{(13)} \end{align*}

Thus the required nonlinear first-order PDE is

\begin{equation*} z^2(p^2+q^2+1)=r^2. \end{equation*}

The two solved sphere problems use the same idea: differentiate first, then remove the arbitrary constants. Choose a sphere family and click Show Elimination to see the differentiated equations and the final PDE. Notice that the one-constant family produces

yp-xq=0

, while the two-constant family produces

z^2(p^2+q^2+1)=r^2

. This helps you compare how the number and position of arbitrary constants affect the elimination step.

All surfaces of revolution about the

z

-axis may be written in the form

\begin{equation*} z=f(x^2+y^2), \quad \text{(14)} \end{equation*}

where

f

is an arbitrary function. Put

\begin{equation*} u=x^2+y^2. \end{equation*}

Then (14) becomes

z=f(u)

. Differentiating with respect to

x

and

y

, we get

\begin{align*} p &=2x f'(u), \nonumber \\ q &=2y f'(u). \quad \text{(15)} \end{align*}

Now multiply the first equation in (15) by

y

and the second equation in (15) by

x

. Then

\begin{align*} yp &=2xy f'(u), \nonumber \\ xq &=2xy f'(u). \quad \text{(16)} \end{align*}

Therefore

\begin{align*} yp-xq &=0. \quad \text{(17)} \end{align*}

Hence surfaces of revolution about the

z

-axis satisfy

\begin{equation*} yp-xq=0. \end{equation*}

A surface of revolution about the

z

-axis has the form

z=f(x^2+y^2)

. Choose a sample function, enter a point, and click Verify Relation to compute

p

q

, and

yp-xq

. Notice that

p

and

q

may change from point to point, but the combination

yp-xq

cancels to zero. This shows why all such surfaces satisfy the same first-order PDE.

When constructing a PDE, do not try to solve for

z

first. Differentiate the given family and then eliminate the arbitrary constants. The final answer should not contain those arbitrary constants.

The equation

yp-xq=0

is linear in

p

and

q

. The equation

z^2(p^2+q^2+1)=r^2

is nonlinear because it contains

p^2

and

q^2

. Thus the same construction method may produce either a linear or a nonlinear first-order PDE.

The construction method may produce a linear or nonlinear first-order PDE. Choose a final PDE and click Classify Result to decide whether it is linear in

p

and

q

. Notice that

yp-xq=0

contains

p

and

q

only to the first power, while

z^2(p^2+q^2+1)=r^2

contains squared derivative terms. This distinction explains the remark about linear and nonlinear results.

BMLabs Mathematics

BMLabs Mathematics

Examples of Constructing First-Order PDEs from Families of Spheres

Examples of Constructing First-Order PDEs from Families of Spheres

Construction Method Flow

Moving Sphere Family Viewer

Sphere Elimination Calculator

Surface of Revolution Check

Result Type Classifier

Practice Constructing First-Order PDEs

Coordinates and Locus

Invariants under Orthogonal Transformation