Question

Solve each differential equation.$$\frac{d y}{d x}=4 x^{2}$$

Short answer

The solution to the differential equation \( \frac{dy}{dx} = 4x^2 \) is \( y = \frac{4}{3}x^3 + C \. where \( C \) is the constant of integration.

Step-by-step solution

Understand the Differential Equation

The given differential equation is of the form \( \frac{dy}{dx} = f(x) \), where \( f(x) \) is a function of \( x \). In this case, \( f(x) = 4x^2 \). It represents how the function \( y \) changes with respect to \( x \) and indicates that the rate of change of \( y \) with respect to \( x \) is given by \( 4x^2 \).

Integrate Both Sides of the Differential Equation

To find the solution of the differential equation, integrate both sides with respect to \( x \). The integral of the right side with respect to \( x \) is the integral of \( 4x^2 \), and the integral of \( \frac{dy}{dx} \) with respect to \( x \) gives us \( y \).

Perform the Integration

The integral of \( 4x^2 \) with respect to \( x \) is \( \frac{4}{3}x^3 + C \), where \( C \) is the constant of integration, which can be any real number. Therefore, the solution to the differential equation is \( y = \frac{4}{3}x^3 + C \).

Key concepts

Integration Techniques

Understanding integration techniques is crucial when solving differential equations because it allows us to find the antiderivatives necessary to solve for the variable of interest. In the context of the given problem, we are faced with integrating a simple monomial, specifically, the term 4x^2. To integrate this expression with respect to x, you use the power rule for integration.

The power rule states that the integral of x^n is (x^(n+1)) / (n+1), plus the constant of integration, where n is any real number other than -1. Applying this rule to 4x^2, you increase the exponent by one to get 3, and divide by this new exponent, resulting in 4x^3/3. This simplifies to (4/3)x^3, plus the constant of integration, represented by C.

It's crucial to understand various integration techniques because different differential equations can require different methods, such as u-substitution, integration by parts, or partial fractions, especially when dealing with more complex functions.

Rate of Change

The rate of change is a fundamental concept in calculus, describing how one quantity changes in relation to another. In our differential equation, dy/dx represents the rate of change of y with respect to x, essentially telling us how y is expected to alter as x varies.

In concrete terms, 4x^2 indicates that the rate at which y changes is proportional to the square of x; as x grows larger, the change in y accelerates accordingly. This relationship is visually represented by the slope of the tangent to the curve at any given point on a graph of y versus x. Understanding the rate of change allows us to predict behavior in dynamic systems and is a fundamental aspect of many fields such as physics, engineering, and economics.

Constant of Integration

The constant of integration is a critical part of solving an indefinite integral. It represents the infinite number of antiderivatives that a function can have since any constant value added to a function does not change the derivative.

When integrating 4x^2, we write the solution as (4/3)x^3 + C, where C is the constant of integration. This C accounts for the fact that there could have been a constant term in the original function y, which disappeared when the function was differentiated to give 4x^2. Without including C, we would be missing an infinite number of potential solutions. In practical applications, the value of C is often determined by an initial condition, a value of y at a specific point of x, allowing for the precise solution to a differential equation. It's this constant that makes the general solution applicable to a multitude of scenarios, each with different starting conditions.