10 Questions
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.
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'.
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.
From the given equations, we can express $p$ as $p = -\frac{x},{z - c}$ and $q$ as $q = -\frac{y},{z - c}$.
What is the equation obtained by substituting the expressions for $p$ and $q$ into the original equation $(x^2)p + ay^2 + b = z?
Substituting the expressions for $p$ and $q$ into the original equation, we obtain the equation $2x(-\frac{y},{z - c}) = 4xyz$.
Obtain the partial differential equation by eliminating the arbitrary constants 'a' and 'b' from the equation $(x^2)p + ay^2 + b = z$.
The partial differential equation obtained by eliminating 'a' and 'b' is $(x^2)p + y^2 = 2xz$.
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 $x^2 + y^2 + (z - c)^2 = r^2$.
Differentiate the equation of the sphere $x^2 + y^2 + (z - c)^2 = r^2$ with respect to 'x' and 'y'.
Differentiating with respect to 'x', we get $2x + 2(z - c)p = 0$, and differentiating with respect to 'y', we get $2y + 2(z - c)q = 0$.
What is the equation obtained by differentiating the sphere equation with respect to 'x'?
The equation obtained by differentiating with respect to 'x' is $2x + 2(z - c)p = 0$.
What is the equation obtained by differentiating the sphere equation with respect to 'y'?
The equation obtained by differentiating with respect to 'y' is $2y + 2(z - c)q = 0.
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
This quiz covers the topic of partial differential equations and how to obtain the equation by eliminating arbitrary constants 'a' and 'b' from a given expression. It will test your understanding of transformations and partial differential equations in mathematics.
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