Problem 41 - Calculus and its applications

Question

Solve by separating variables. \(\frac{d y}{d x}=\frac{6}{y}\)

Step-by-Step Solution

Answer

The solution is \( y = \pm \sqrt{12x + C} \).

Step 1: Separate Variables

Start by rewriting the equation in a form that allows you to separate the variables. Multiply both sides by \(y\) and \(dx\) to get: \[ y \, dy = 6 \, dx \] Now, the variables are separated, with all \(y\) terms on one side and \(x\) terms on the other.

Step 2: Integrate Both Sides

Integrate both sides of the equation with respect to their respective variables.\[ \int y \, dy = \int 6 \, dx \]The left side integrates to \(\frac{y^2}{2}\) and the right side integrates to \(6x + C\), where \(C\) is the constant of integration.

Step 3: Solve for Constants and Rearrange

After integration, you will have:\[ \frac{y^2}{2} = 6x + C \]Multiply the whole equation by 2 to simplify it:\[ y^2 = 12x + 2C \]Let \(C' = 2C\), so we rewrite the equation as:\[ y^2 = 12x + C' \]This can then be rearranged to solve for \(y\), if needed:\[ y = \pm \sqrt{12x + C'} \]

Step 4: Write the General Solution

The final solution expresses \(y\) in terms of \(x\):\[ y = \pm \sqrt{12x + C'} \]This equation represents the family of curves that satisfy the original differential equation.

Key Concepts

Separation of Variables

When dealing with differential equations, the method of separation of variables is a powerful tool. It's especially effective for equations where the variables can be independently manipulated. In the given exercise, we have the equation \( \frac{dy}{dx} = \frac{6}{y} \). We separate the variables by performing algebraic manipulations to isolate the differentials of \( y \) and \( x \) on opposite sides of the equation.

Here's how it works:

• Multiply both sides by \( y \) to remove the \( y \) in the denominator from the right side and by \( dx \) to transfer the derivative part to the other side: \( y \, dy = 6 \, dx \).
• This results in having both \( y \) and \( dy \) on one side and \( x \) and \( dx \) on the other. Now each variable is neatly organized, making the equation ready for integration.

By successfully separating the variables, we're setting the groundwork to integrate each side independently, which is the next step in solving the equation.

Integration Techniques

Integration allows us to find functions from their derivatives. After separation of variables, the next logical step is to integrate both sides. This will help in finding the original function.

• On the left-hand side, integrate \( \int y \, dy \). The antiderivative of \( y \) with respect to \( y \) is \( \frac{y^2}{2} \).
• On the right-hand side, integrate \( \int 6 \, dx \). The antiderivative is a linear term: \( 6x \).

Don’t forget the constant of integration, \( C \), which is essential because it represents the family of all possible solutions.

So, after integration, you have the equation: \( \frac{y^2}{2} = 6x + C \).
To make it simpler, multiply the entire equation by 2, leading to: \( y^2 = 12x + 2C \). For convenience, we can rename \( 2C \) as \( C' \), resulting in: \( y^2 = 12x + C' \). This simplification is crucial for understanding the structure of the solution and lays the foundation for finding the general solution of the equation.

General Solution

The end goal of solving a differential equation is to obtain a general solution that describes the relationship between the variables. In this problem, the last expression we found was \( y^2 = 12x + C' \). From this, we can derive the general solution.

• First, solve for \( y \) by taking the square root of both sides: \( y = \pm \sqrt{12x + C'} \).
• The \( \pm \) symbol indicates that there are typically two solutions for each value of \( x \): one positive and one negative. This represents a family of curves, showcasing the range of shapes and behaviors the solutions might take depending on the value of \( C' \).

This expression, \( y = \pm \sqrt{12x + C'} \), encompasses the entire family's possible solutions to our original equation.

Each \( C' \) value represents a distinct curve or function satisfying the differential equation \( \frac{dy}{dx} = \frac{6}{y} \). This carefully crafted structure allows us to explore a wide variety of possibilities that fit within the parameters provided by the original problem.