Problem 18 - Advanced Engineering Mathematics

Question

Identifying \(A=3, B=5,\) and \(C=1,\) we compute \(B^{2}-4 A C=13 > 0 .\) The equation is hyperbolic.

Step-by-Step Solution

Answer

The equation is hyperbolic because the discriminant (13) is greater than zero.

Step 1: Identify Values

First, we identify the values given in the problem: - \( A = 3 \) - \( B = 5 \) - \( C = 1 \).

Step 2: Compute Discriminant

Next, we calculate the discriminant using the formula \( B^{2} - 4AC \). Substitute the given values into the formula:- \( B^{2} = 5^{2} = 25 \)- \( 4AC = 4 \times 3 \times 1 = 12 \)- \( B^{2} - 4AC = 25 - 12 = 13 \).

Step 3: Analyze Discriminant

Determine the classification of the equation based on the discriminant:- The discriminant \( 13 \) is greater than zero, which means the equation is hyperbolic. This conclusion follows from the property that when \( B^{2} - 4AC > 0 \), the conic section represented by the equation is hyperbolic.

Key Concepts

Discriminant

The discriminant is a crucial component in differential equations and polynomial equations. It is used to determine the nature of solutions or roots of the equation. For conic sections, its calculation is based on the formula \( B^{2} - 4AC \). This formula characterizes the type of conic section represented by the general equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).
Here’s how it works:

• If \( B^{2} - 4AC > 0 \), the conic section is hyperbolic, indicating that the graph is a pair of hyperbolas.
• If \( B^{2} - 4AC = 0 \), the conic section is parabolic, resulting in a parabolic graph.
• If \( B^{2} - 4AC < 0 \), the conic section is elliptic, forming an ellipse or a circle.

The discriminant provides a mathematical way to predict shape and intersection patterns of conics, which is crucial when solving related differential equations.

Hyperbolic Equation

A hyperbolic equation is characterized by the discriminant being greater than zero. In terms of conic sections, it describes a hyperbola. Hyperbolas open towards infinity and consist of two separate curves called branches. These curves are shaped like 'mirror images' and have two vertices and two foci which define their spatial properties.
In our case, the discriminant is \(13\) which is greater than zero, confirming the hyperbolic nature of the equation. Key properties include:

• Two branches that open either horizontally or vertically.
• Associative with real symmetric matrices with positive eigenvalues.
• Asymptotes that cross through the center of the hyperbola, providing a guideline for its branch paths.

These properties are fundamental when dealing with systems of differential equations where solutions might take hyperbolic forms.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The resulting curves — parabolas, circles, ellipses, and hyperbolas — are integral in fields such as physics, engineering, and astronomy. Each type of curve depends on the angle of the intersecting plane relative to the cone’s base.Differential equations frequently involve conic sections, where the solution paths might align with these geometrical shapes.
Here are the types of conic sections based on the discriminant:

• Ellipse: Occurs when \( B^{2} - 4AC < 0 \) and includes circular shapes as special cases.
• Parabola: Forms when \( B^{2} - 4AC = 0 \), showing a symmetric open curve.
• Hyperbola: Represented by \( B^{2} - 4AC > 0 \) with two separate branches.

Understanding conic sections involves visualizing how changes in equation coefficients affect their shape and orientation.