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Complex Plane

PDE

Essence of Math

Toolness Of Math

The Number Line

Fractions

Negative Numbers

Irrational Numbers

Variables

Equations

Functions

Modulus

Real Numbers

Vector Spaces

Binary Numbers

Complex Plane

PDE

**Problem 1**: what are conic sections or conics?
**Ans: ** consider the double cone of figure 1. Some of the cross sections (plane slices) that can be obtained from the double cone are:

(a) Single point

(b) Single line

(c) Double lines

(d) Circle

(e) Parabola

(f) Ellipse

(g) Hyperbola

These shapes are called conic sections or conics. It is shown in the ellipse problems that the circle is a special case of an ellipse. The parabola, ellipse and hyperbola are called *non-degenerate conics* while the single point, single line and double lines are called *degenerate conics*

**Problem 2**: the standard forms of the equations of non-degenerate conics are different because the values of their respective eccentricities are different. What is meant by the *eccentricity* of a *non-degenerate* conic?
** Ans: **The *eccentricity* of a *non-degenerate* conic is a constant (denoted by *e*) that is incorporated into its definition. In general, *non-degenerate* conics are defined as *the set of points P in R ^{2} that satisfy the following distance equality*:

Where PF is the distance from P to a fixed point F called the

A

**Problem 3**: Write a general equation that can represent the equations of non-degenerate conics.
**Ans: ****Ax ^{2} + Bxy + Cy^{2} + Fx + Gy + H = 0**------(2). Where A, B and C are real numbers and are not all zero.

**Problem 4 **: given the general equation of conics:

**Ax ^{2} + Bxy + Cy^{2} + Fx + Gy + H = 0**

and the associated matrices of figure 2.

(a) Write the general equation of conics in terms of the matrices

(b) How is the determinant of matrix

(b) If det

If det

If det

Another way of stating the determinant test for identifying conics is as follows:

If B

If B

If B

**Problem 5 **: identify the following conics:

(a) 9x^{2} - 24xy + 6y^{2} - 6x + 12y - 48 = 0.

(b) 2x^{2} - 16xy + 32y^{2} - 2x + 16y - 24 = 0.

(c) 13x^{2} - 18xy + 19y^{2} - 8x + 18y - 28 = 0.
**Ans:**(a) A = 9, B = -24, C = 6

Therefore, B^{2} - 4AC = 576 - 216 = 360 > 0. Therefore conic is a hyperbola.

(b) A = 2, B = -16, C = 32.

Therefore, B^{2} - 4AC = 256 - 256 = 0. Therefore conic is a parabola.

(c) A = 13, B = -18, C = 19

Therefore, B^{2} - 4AC = 324 - 988 = -664 < 0. Therefore conic is an ellipse.

**Problem 6**: (a) can partial differential equations (PDEs) be expressed as conics?

(b) Write the general form for partial differential equations that are conics.

(c) What constraint is placed on conic partial differential equations?

**Ans:** (a) Yes.

(b) **Au _{xx} + Bu_{xy} + Cu_{yy} + Du_{x} + Eu_{y} + Fu = G**

(c) The partial differential equations must be linear PDEs. That is, the dependent variable u and all its derivatives appear in a linear form in the eqaution (they are not multiplied together or squared).

**Problem 7**: what type of conic is the following PDE?

u_{t} = u_{xx}
**Ans: ** A = 1, B = 0, C = 0.
Therefore, B^{2} - 4AC = 0. Therefore, PDE is parabolic.

**Problem 8**: what type of conic is the following PDE?

u_{xx} + u_{yy} = 0.
**Ans: ** A = 1, C = 1, B = 0.

Therefore, B^{2} - 4AC = -4 < 0. Therefore PDE is elliptic.

**Problem 9**: what type of conic is the following PDE?

u_{xy} = 0
**Ans: **A = 0, B = 1, C = 0.

Therefore, B^{2} - 4AC = 1 > 0. Therefore PDE is hyperbolic.

**Problem 10**: what type of conic is the following PDE

yu_{xx} + u_{yy} = 0
**Ans: ** No help here :).

Peter Oye Sagay