Problem 33 - Advanced Engineering Mathematics

Question

(a) Verify that \(3 x^{2}-y^{2}=c\) is a one-parameter family of solutions of the differential equation \(y d y / d x=3 x\). (b) By hand, sketch the graph of the implicit solution \(3 x^{2}-y^{2}=3 .\) Find all explicit solutions \(y=\phi(x)\) of the DE in part (a) defined by this relation. Give the interval \(I\) of definition of each explicit solution. (c) The point \((-2,3)\) is on the graph of \(3 x^{2}-y^{2}=3\), but which of the explicit solutions in part (b) satisfies \(y(-2)=3 ?\)

Step-by-Step Solution

Answer

(a) Verified; (b) Solutions: \(y = \pm\sqrt{3x^2 - 3}\), \(I = (-\infty,-1] \cup [1,\infty)\); (c) Solution: \(y = \sqrt{3x^2-3}\).

Step 1: Differentiate the given family of solutions implicitly

Start with the given equation: \[ 3x^2 - y^2 = c \]Differentiate both sides with respect to \(x\):- Left side: \( \frac{d}{dx}(3x^2 - y^2) = 6x - 2y \cdot \frac{dy}{dx} \) - Right side: \( \frac{d}{dx}(c) = 0 \) (since \(c\) is constant)Equate and rearrange the terms:\[ 6x - 2y \frac{dy}{dx} = 0 \]\[ 6x = 2y \frac{dy}{dx} \]Solve for \( \frac{dy}{dx} \):\[ \frac{dy}{dx} = \frac{3x}{y} \]Thus, the expression matches the differential equation \(y \frac{dy}{dx} = 3x\).

Step 2: Verify the given function

The function \(3x^2 - y^2 = c\) gives \(\frac{dy}{dx} = \frac{3x}{y}\). Substituting it back into \(y \frac{dy}{dx} = 3x\), we find the equations are consistent with each other. Thus, \(3x^2 - y^2 = c\) is indeed a one-parameter family of solutions.

Step 3: Sketch the curve

Given \(3x^2 - y^2 = 3\), rearrange to express it in a more recognizable form: \[ \frac{x^2}{1} - \frac{y^2}{3} = 1 \]This is the equation of a hyperbola centered at the origin, opening horizontally. Sketching this involves drawing two asymptotes through the origin with slopes \(\pm\sqrt{3}\), and sketching the two branches of the hyperbola along these lines.

Step 4: Solve for explicit solutions

From the equation \(3x^2 - y^2 = 3\), we solve for \(y\):\[ y^2 = 3x^2 - 3 \]\[ y = \pm\sqrt{3x^2 - 3} \]The two explicit solutions \(y = \sqrt{3x^2 - 3}\) and \(y = -\sqrt{3x^2 - 3}\) exist where \(3x^2 - 3 \ge 0\), or \(|x| \ge 1\).Thus, the intervals for the solutions are:\( y = \sqrt{3x^2 - 3} \) is defined for \(x \in (-\infty, -1] \cup [1, \infty) \), and \( y = -\sqrt{3x^2 - 3} \) is defined for the same intervals.

Step 5: Determine specific solution satisfying initial condition

The condition is \(y(-2) = 3\). Substitute \(x = -2\) into the explicit solutions:- For \(y = \sqrt{3x^2 - 3}\): \[ y(-2) = \sqrt{3(-2)^2 - 3} = \sqrt{12 - 3} = \sqrt{9} = 3 \]- For \(y = -\sqrt{3x^2 - 3}\): \[ y(-2) = -\sqrt{12 - 3} = -\sqrt{9} = -3 \]The solution that satisfies the condition \(y(-2) = 3\) is \(y = \sqrt{3x^2 - 3}\).

Key Concepts

Implicit Differentiation

Implicit Differentiation is a method used to find the derivative of a function when it isn't explicitly stated in terms of a single variable. In exercises like the original one, you often encounter equations that involve variables on both sides, such as \(3x^2 - y^2 = c\). Here, differentiating directly with respect to \(x\) isn't straightforward because \(y\) is also part of the equation.
To tackle this, you'd differentiate each part of the equation concerning \(x\), treating \(y\) as a function of \(x\). For instance, when differentiating \(y^2\), consider \(y\) a function of \(x\) and use the chain rule, resulting in \(-2y \cdot \frac{dy}{dx}\). The left side's differentiation gives us \(6x - 2y \cdot \frac{dy}{dx}\), while the right side becomes zero since it's a constant. Next, rearrange to solve for \(\frac{dy}{dx}\), which helps verify if the derived expression aligns with any given differential equation.

Family of Solutions

A Family of Solutions in differential equations comprises all possible solutions that an equation can have based on a parameter. For example, in this context, the term \(3x^2 - y^2 = c\) represents a one-parameter family of solutions for the differential equation \( y \frac{dy}{dx} = 3x \).
The parameter, denoted here as \(c\), can take various values, impacting the specific solution's form. Each value of \(c\) yields a different curve or graph representing a solution belonging to this family. These families allow us to explore general solutions that could satisfy a differential equation under varied conditions.

• Each curve in this family is a hyperbola.
• The value of \(c\) determines the exact form and position of the hyperbola on the graph.
• This family concept is crucial for understanding the range of functions that can satisfy the initial differential equation under different conditions.

Explicit Solutions

Explicit Solutions for a differential equation are those in which the dependent variable is isolated on one side of the equation. In the provided problem, the equation \(3x^2 - y^2 = 3\) can be expressed in such a way that \(y\) is isolated, giving explicit solutions.
By rearranging, the expression turns into \(y = \pm\sqrt{3x^2 - 3}\), providing two potential solutions. These solutions clearly state \(y\) in terms of \(x\) without any implicit relation between them.

• \(y = \sqrt{3x^2 - 3}\) and \(y = -\sqrt{3x^2 - 3}\) cover all scenarios where the original equation holds true.
• Being explicit makes it easier to graph these solutions and analyze them under specified conditions.
• The explicit approach simplifies identifying the values of \(y\) for given values of \(x\).

Hyperbola

Understanding a Hyperbola is crucial when dealing with equations like \(3x^2 - y^2 = c\), which simplifies into a hyperbolic equation. A hyperbola consists of two separate curves or branches that mirror each other, characterized by their opening direction and asymptotes.
In its standard form, \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), as seen with \(\frac{x^2}{1} - \frac{y^2}{3} = 1\), the hyperbola opens horizontally. Sketching this requires:

• Determining the slopes of the asymptotes, here \(\pm\sqrt{3}\), which guide the shape.
• Identifying branches in each quadrant symmetry.
• Marking the points of intersection with the axes to highlight where curves approach but never touch.

The hyperbola helps visualize and interpret the range of a family of solutions, providing insights into how they might behave geometrically.

Intervals of Definition

Intervals of Definition indicate the range of \(x\) values where a function or solution is valid and real. For explicit solutions derived from the given hyperbolic equation, it’s vital to set these intervals.
Taking \(y = \pm\sqrt{3x^2 - 3}\), the expression inside the square root determines the domain of \(x\). The inequality \(3x^2 - 3 ≥ 0\) simplifies to \(|x| ≥ 1\), establishing intervals for valid solutions.

• Both solutions, \(y = \sqrt{3x^2 - 3}\) and \(y = -\sqrt{3x^2 - 3}\), are defined for \(x \in (-\infty, -1] \cup [1, \infty)\).
• Defining these intervals accurately ensures we understand where the functions exist without producing imaginary numbers.
• This helps accurately answer questions involving values of solutions at specific points, such as verifying \(y(-2) = 3\).