Problem 1 - The Calculus with Analytic Geometry

Question

\frac{d y}{d x}=3 x^{2}+2 x-7

Step-by-Step Solution

Answer

The solution is \(y = x^3 + x^2 - 7x + C\).

Step 1: Identify the integral to solve

Given the differential equation \(\frac{dy}{dx} = 3x^2 + 2x - 7\), recognize that the solution involves finding the antiderivative or integral of the right-hand side.

Step 2: Integrate each term

Integrate the right-hand side term-by-term: \(\frac{dy}{dx} = 3x^2 + 2x - 7\). The integral of \(3x^2\) is \(x^3\), the integral of \(2x\) is \(x^2\), and the integral of \(-7\) is \(-7x\).

Step 3: Include the constant of integration

Combine the integrals of each term to form the solution to the differential equation. Don't forget to include the constant of integration \(C\). This gives: \(\frac{dy}{dx} = 3x^2 + 2x - 7\) integrates to \(y = x^3 + x^2 - 7x + C\).

Key Concepts

Differential Equations

Differential equations are equations that involve an unknown function and its derivatives. In simpler terms, they describe how a function changes over time or according to another variable. Understanding differential equations is extremely important in many fields, such as physics, engineering, and economics. The exercise involves solving the differential equation \(\frac{dy}{dx} = 3x^2 + 2x - 7\). To solve it, we need to find the function y that satisfies this equation, indicating how y changes with respect to x.

Antiderivative

The antiderivative, also known as the indefinite integral, is essentially the reverse process of differentiation. When you differentiate a function, you are finding how it changes, but when you integrate it, you are finding the original function from its rate of change. In the given exercise, to solve the differential equation \(\frac{dy}{dx} = 3x^2 + 2x - 7\), we find the antiderivative of \(3x^2 + 2x - 7\). This process involves integrating each term separately:

• Integral of \(3x^2\) is \(x^3\)
• Integral of \(2x\) is \(x^2\)
• Integral of \(-7\) is \(-7x\)

Combining these results, the antiderivative is \(x^3 + x^2 - 7x\). But there’s one more important thing we need to remember...

Constant of Integration

When you find an antiderivative, you must include a constant of integration, often denoted as \(C\). This constant represents an infinite number of possible functions that differ only by a constant value. The reason for this is that when you differentiate a constant, the result is 0, meaning any constant could have been there originally. In our exercise, combining the antiderivatives and including the constant, we get the solution to the differential equation: \(\frac{dy}{dx} = 3x^2 + 2x - 7\) integrates to \(y = x^3 + x^2 - 7x + C\). Always remember the constant of integration, as it can change the general solution significantly.