Question

Finding a General Solution Using Separation of Variables In Exercises \(1-14,\) find the general solution of the differential equation. $$ \sqrt{x^{2}-16} y^{\prime}=11 x $$

Short answer

The general solution to the differential equation is \(\frac{1}{2}y^2 = 11\sqrt{x^2 - 16} + C\).

Step-by-step solution

Rewrite the differential equation

Rewrite the given differential equation to isolate \(y'\). This yields \(y' = \frac{11x}{\sqrt{x^2 - 16}}\).

Implement separation of variables

Next, rewrite the equation in differential form and apply separation of variables. The equation becomes \(y' dx = \frac{11x}{\sqrt{x^2 - 16}} dx\), which can be rewritten as \(y dy = \frac{11x}{\sqrt{x^2 - 16}} dx\).

Integration

Integrate both sides of the equation. The left side integrates to \(\frac{1}{2}y^2\). The right side integrates using the substitution method - let \(u = x^2 - 16, du = 2x dx\), so the integral becomes \(\frac{11}{2}\int \frac{du}{\sqrt{u}}\), which simplifies to \(\frac{11}{2} \cdot 2\sqrt{u} = 11\sqrt{u}\). Substituting back for \(u\), the integral becomes \(11\sqrt{x^2 - 16}\).

General solution

Combining the two integrals gives the general solution to the differential equation: \(\frac{1}{2}y^2 = 11\sqrt{x^2 - 16} + C\), where \(C\) is the constant of integration.

Key concepts

Differential Equation

A differential equation is a type of mathematical equation that involves functions and their derivatives. In simple terms, it describes how a function changes over time. Imagine you have a graph of a line or a curve. The slope (or steepness) at each point of this graph may vary, and a differential equation helps us understand this changing slope.
In our exercise, the given differential equation is \( \sqrt{x^2 - 16} y' = 11x \).
Here, \( y' \) represents the derivative of the function \( y \) concerning \( x \). This particular type of differential equation is known as a first-order differential equation because it includes only the first derivative (\( y' \)).

• First-Order: Involves the first derivative \( y' \).
• Non-Linear: Includes terms that are not simply \( y \) or \( y' \) but other combinations, like multiplying by a square root of an expression.

By solving this equation, we can find a relationship or "rule" that describes the original function \( y \).

Integration

Integration is the process of finding the original function from its derivative, a reverse operation of differentiation. Think of it as piecing together the puzzle from small pieces back to the whole picture. This technique is essential for solving differential equations, which require us to restore the function from its rate of change (derivative).
In our exercise, we arrived at an equation in the form of an integral: \( y \, dy = \frac{11x}{\sqrt{x^2 - 16}} \cdot dx \). We need to integrate both sides.

• Left Side: Integrating \( y \cdot dy \) gives \( \frac{1}{2}y^2 \).
• Right Side: Requires substitution to integrate properly.

This integration process involves calculating the antiderivative, which basically means finding a function whose derivative is the given integrand (the function to be integrated).

Substitution Method

The substitution method is a clever technique used in calculus to simplify the process of integration. It's like replacing a complex part of a puzzle with a simpler part to make it easier to solve.
In this exercise, we employ substitution to handle the complicated integral on the right side of the equation.
We let \( u = x^2 - 16 \), which means \( du = 2x \, dx \). Here's a quick step-by-step of the process:

• Substitute \( x^2 - 16 \) with \( u \), simplifying the expression under the square root.
• The differential \( dx \) changes to reflect \( du \), giving \( du = 2x \cdot dx \).
• This allows the integral \( \frac{11x}{\sqrt{x^2 - 16}} \cdot dx \) to be rewritten as \( \frac{11}{2} \int \frac{du}{\sqrt{u}} \).

This simplification makes the integral on the right-hand side easier to solve, leading us to a general solution.

General Solution

Once we have integrated both sides of a differential equation, the result is what we call the general solution. It represents not just one single function, but a family of functions that satisfy the original differential equation.
In this exercise, after integrating, we found: \( \frac{1}{2}y^2 = 11\sqrt{x^2 - 16} + C \). Here, \( C \) is a constant of integration that accounts for the initial conditions or specific values of the problem.

• Constant \( C \): Represents any constant number that can adjust the function based on initial data.
• Represents a range or family of curves rather than a single curve.

Having a general solution is like having a guidebook—it's not just one path but all possible paths that align with the given rate of change expressed by the differential equation.