Question

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

Short answer

By finding the derivative \(Dy = 2Cx\) of the proposed solution \(y = Cx^2\) and verifying that it satisfies the given differential equation \(Dy = \frac{2y}{x}\), we show that \(y = Cx^2\) is a solution to the differential equation.

Step-by-step solution

Identify the differential equation and the solution

First, identify the given differential equation which is expressed as the derivative of y with respect to x, written as \(Dy = \frac{2y}{x}\), and the proposed solution \(y = Cx^2\), where \(C\) is an arbitrary constant.

Find the derivative of the proposed solution

Next, we take the derivative of the proposed solution \(y = Cx^2\) with respect to \(x\) to find \(Dy\). The derivative of \(Cx^2\) using the power rule is \(Dy = \frac{d(Cx^2)}{dx} = 2Cx\).

Substitute the derivative into the differential equation

Now, we substitute the derivative \(Dy = 2Cx\) into the left side of the given differential equation and see if it equals the right side, which is \(\frac{2y}{x}\).

Verify the equality

Substitute \(y = Cx^2\) into \(\frac{2y}{x}\) to get \(\frac{2Cx^2}{x} = 2Cx\). Since \(2Cx\) is identical to our expression for \(Dy\), the proposed solution \(y = Cx^2\) is indeed a solution to the differential equation \(Dy = \frac{2y}{x}\).

Key concepts

Verifying Solutions to Differential Equations

Understanding how to check if a given function is a solution to a differential equation is a fundamental skill in calculus and applied mathematics. To verify a solution, we essentially work backwards, substituting the solution into the original differential equation to see if it results in an identity - that is, a true statement for all values within the domain of the variable.

Let's consider the process using the exercise provided. We are given a differential equation, specifically a first-order equation, where the rate of change of the function y (denoted as \(Dy\)) is proportional to the function itself scaled by a factor varying inversely with the variable x. The first step involves identifying the original equation and recognizing the structure of the purported solution. In this case, the solution's structure, \(y = Cx^2\), must comply with the proportional relationship stated in \(Dy = \frac{2y}{x}\).

In the verification process, we are calculating the derivative of the proposed solution to see if it matches the differential equation's right-hand side when substituted back. If after substituting we find that each side of the equation is equivalent, our proposed solution is indeed correct. This methodology is universal for verifying solutions to differential equations and is an essential part of differential calculus.

Derivative Calculation

To analyze changes in functions, calculus introduces the concept of derivatives. The ability to compute derivatives is crucial to solving many problems involving rates of change, such as those found in physics, economics, and biology. In the context of our exercise, after identifying the differential equation and proposed solution, we proceed with finding the derivative of the function \(y = Cx^2\).

The derivative represents the slope of the tangent line to the function at any given point and mathematically describes how the function values change with respect to changes in the independent variable, in this case, x. Calculating the derivative involves applying differentiation rules which are designed to make this process systematic and efficient. In practice, like we see here, the derivative of a solution must align with the behavior described by the differential equation for the solution to be correct.

Power Rule for Differentiation

The power rule is one of the most fundamental and widely-used rules for differentiation. It simplifies the process of finding the derivative for monomials - that is, single-term algebraic expressions in which a constant is multiplied by the variable raised to a power. The rule states that if you have a function \(y = ax^n\), then its derivative with respect to x is \(Dy = anx^{n-1}\).

This rule is used in the second step of the provided solution when calculating the derivative of the function \(y = Cx^2\). By applying the power rule, we easily find that the derivative is \(Dy = 2Cx\). The simplicity of the power rule makes it a critical tool for students and professionals alike in the fields requiring calculus. Decomposing more complex functions into parts that can be differentiated using the power rule is a common strategy in calculus for tackling a variety of problems. This step in the differentiation process is vital to verify if the proposed function is, in fact, a solution to the original differential equation.