Question

Show that each function is a solution to the given differential equation. $$y^{\prime}=\frac{2 y}{x}, y=C x^{2}$$

Short answer

Taking the derivative of the function \(y = C x^2\) yields \(y' = 2C x\). Substituting this into the differential equation \(y' = \frac{2y}{x}\) and simplifying results in an identity, confirming that \(y = C x^2\) is a solution.

Step-by-step solution

Express the given function

We are given the function in the form of an equation, where the value of the function is given by the equation: \(y = C x^2\). This represents a family of curves, where C is a constant that could have any value.

Differentiate the function with respect to x

The differential equation requires the first derivative of y with respect to x. Differentiate \(y = Cx^2\) using the power rule, which states that the derivative of \(x^n\) with respect to x is \(nx^{n-1}\). Through differentiation, we get \(y' = d(Cx^2)/dx = 2C x^{2-1} = 2C x\).

Substitute the derivative in the differential equation

We substitute the derived value of \(y'\) into the given differential equation \(y' = \frac{2y}{x}\). Our calculated derivative is \(2C x\), and our expression for y is \(C x^2\). Plugging these in, we get \(2C x = \frac{2(C x^2)}{x}\).

Simplify the equation

Simplify the equation from the previous step. The x's on the right side of the equation cancel out one from the numerator and denominator, giving us \(2C x = 2C x\). Since both sides of the equation are equal, this confirms that the function \(y = C x^2\) is indeed a solution to the differential equation.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus, referring to the process of finding the derivative of a function. In essence, differentiation measures how a function changes as its input changes. The derivative represents the rate of this change, often thought of as the slope of the tangent line to the graph of the function at any given point.

To differentiate a function means to determine this rate of change mathematically. In the context of the exercise, we are concerned with differentiating the function with respect to a single variable, x. This process is crucial as it allows us to understand how the output of the function, y, responds to changes in x. When we differentiate a function like \( y = C x^2 \), we are essentially looking for a new function, \( y' \), that gives us the slope at any point along the curve described by the original function.

Power Rule

The power rule is a basic rule of differentiation that is essential for finding the derivative of functions that are polynomial in form. It's one of the simplest and most widely used rules, making it an important tool in calculus.

The power rule states that to find the derivative of a term like \( x^n \), where n is any real number, you multiply by the exponent n and then reduce the exponent by one to get \( nx^{n-1} \). In the given exercise, we apply the power rule to the function \( y = C x^2 \), treating C as a constant multiplier.

Applying the power rule gives us the first derivative, \( y' \), as \( 2C x^{2-1} \) which simplifies to \( 2C x \). This step transforms the original function into its rate of change counterpart, which is critical for solving differential equations and understanding the behavior of the function.

First Derivative

The first derivative of a function provides valuable information about the function's behavior, indicating how it changes at any point in its domain. When we differentiate a function once, the result is called the first derivative, often denoted as \( f'(x) \) or \( y' \) when the function is given as y.

In the case of our differential equation \( y' = \frac{2y}{x} \), the first derivative represents the instantaneous rate of change of y with respect to x. By computing the first derivative of \( y = C x^2 \), as shown in the previous steps using the power rule, we have calculated that \( y' = 2C x \). This allows us to make a direct comparison with the given differential equation and determine whether our function is indeed a solution.

Here the first derivative, which is linear and directly proportional to x, fits perfectly into the equation provided, completing the solution process and establishing that \( y = C x^2 \) satisfies the differential equation.