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Obtain the partial differential equation by eliminating the arbitrary constants 'a' and 'b' from the given equation $(x^2)p + ay^2 + b = z$.
Obtain the partial differential equation by eliminating the arbitrary constants 'a' and 'b' from the given equation $(x^2)p + ay^2 + b = z$.
The partial differential equation can be obtained by differentiating the given equation partially with respect to 'x' and 'y', resulting in the equations $(x^2)p + b = 2xq$ and $ay^2 + b = 2yp$. Substituting these back into the original equation, we get $(2xq)(2yp) = 4xyz$, which is the partial differential equation.
Form the partial differential equation of all spheres whose centers lie on the z-axis.
Form the partial differential equation of all spheres whose centers lie on the z-axis.
The partial differential equation of all spheres whose centers lie on the z-axis is $x^2 + y^2 + (z - c)^2 = r^2$, where $c$ is the constant representing the center of the sphere and $r$ is the radius.
Differentiate the equation $x^2 + y^2 + (z - c)^2 = r^2$ with respect to 'x' and 'y'.
Differentiate the equation $x^2 + y^2 + (z - c)^2 = r^2$ with respect to 'x' and 'y'.
When differentiated with respect to 'x', we get $2x + 2(z - c)p = 0$, and when differentiated with respect to 'y', we get $2y + 2(z - c)q = 0$.
Express the equations $x = -(z - c)p$ and $y = -(z - c)q$ in terms of $p$ and $q.
Express the equations $x = -(z - c)p$ and $y = -(z - c)q$ in terms of $p$ and $q.
What is the equation obtained by substituting the expressions for $p$ and $q$ into the original equation $(x^2)p + ay^2 + b = z?
What is the equation obtained by substituting the expressions for $p$ and $q$ into the original equation $(x^2)p + ay^2 + b = z?
Obtain the partial differential equation by eliminating the arbitrary constants 'a' and 'b' from the equation $(x^2)p + ay^2 + b = z$.
Obtain the partial differential equation by eliminating the arbitrary constants 'a' and 'b' from the equation $(x^2)p + ay^2 + b = z$.
Form the partial differential equation of all spheres whose centers lie on the z-axis.
Form the partial differential equation of all spheres whose centers lie on the z-axis.
Differentiate the equation of the sphere $x^2 + y^2 + (z - c)^2 = r^2$ with respect to 'x' and 'y'.
Differentiate the equation of the sphere $x^2 + y^2 + (z - c)^2 = r^2$ with respect to 'x' and 'y'.
What is the equation obtained by differentiating the sphere equation with respect to 'x'?
What is the equation obtained by differentiating the sphere equation with respect to 'x'?
What is the equation obtained by differentiating the sphere equation with respect to 'y'?
What is the equation obtained by differentiating the sphere equation with respect to 'y'?
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Study Notes
Partial Differential Equation of a Sphere
- Given equation: $(x^2)p + ay^2 + b = z$, where 'a' and 'b' are arbitrary constants
- To obtain the partial differential equation, eliminate 'a' and 'b' from the equation
- The equation represents a sphere with center on the z-axis
Sphere Equation
- The equation of a sphere is: $x^2 + y^2 + (z - c)^2 = r^2$, where 'c' is the center of the sphere and 'r' is the radius
- Differentiating the sphere equation with respect to 'x' gives: $2x + 2(z - c)(-p) = 0$, where $p = \frac{\partial z}{\partial x}$
- Differentiating the sphere equation with respect to 'y' gives: $2y + 2(z - c)(-q) = 0$, where $q = \frac{\partial z}{\partial y}$
- Expressing the equations in terms of 'p' and 'q' gives: $x = -(z - c)p$ and $y = -(z - c)q$
- Substituting these expressions into the original equation gives: $(x^2)p + ay^2 + b = z$, where 'a' and 'b' are eliminated
- The resulting partial differential equation represents all spheres whose centers lie on the z-axis
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