Question

Finding general solutions Find the general solution of each differential equation. Use \(C, C_{1}, C_{2}, \ldots\) to denote arbitrary constants. $$p^{\prime}(x)=\frac{16}{x^{9}}-5+14 x^{6}$$

Short answer

Question: Find the general solution of the first-order differential equation: \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^6\). Answer: The general solution of the first-order differential equation is: \(p(x) = -2x^{-8} - 5x + 2x^7 + C\), where C is an arbitrary constant.

Step-by-step solution

Identify the type of differential equation

The given differential equation is a first-order differential equation, as it is expressed in terms of the first derivative \(p'(x)\).

Rewrite the equation

The given differential equation is: $$p'(x) = \frac{16}{x^{9}} - 5 + 14x^6$$

Integrate both sides with respect to x

To find the general solution, we need to integrate the given equation with respect to x: $$p(x) = \int p'(x) dx$$

Integrate the given function

Integrate each term on the right side of the equation: $$p(x) = \int (\frac{16}{x^{9}} - 5 + 14x^6) dx$$ Break the integral into the sum of three integrals: $$p(x) = \int \frac{16}{x^{9}} dx - \int 5 dx + \int 14x^6 dx$$

Apply the integration rules

Apply the power rule for integration for each term: $$p(x) = 16\int x^{-9} dx - 5\int 1 dx + 14\int x^6 dx$$ For each term, add 1 to the power and divide by the resulting power: $$p(x) = 16\frac{x^{-8}}{-8} - 5\frac{x^1}{1} + 14\frac{x^7}{7} + C$$

Simplify the results

Simplify the resulting integral expression and include the constant \(C\): $$p(x) = -2x^{-8} - 5x + 2x^7 + C$$

Write the general solution

The general solution of the first-order differential equation is: $$p(x) = -2x^{-8} - 5x + 2x^7 + C$$ Include the arbitrary constant \(C\) in the general solution.

Key concepts

Understanding Integration in Differential Equations

Integration is a mathematical technique used to find the antiderivative of a function. In the context of differential equations, particularly first-order differential equations, integration allows us to reverse the process of differentiation. This is key when tasked with finding a general solution. When we are given a derivative, like in our exercise with the equation \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^6\), we integrate this to find \(p(x)\), the original function.Here's how you can think about integration in simple steps:

• Recognize the function you need to integrate.
• Use known integration rules, such as the power rule, which states \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), provided \(n eq -1\).
• Apply these rules to each term in the equation separately.

The integration process not only yields the function \(p(x)\) but also introduces an arbitrary constant 'C'. This constant represents a family of solutions, reflecting different initial conditions the system might have.

What is the General Solution?

In differential equations, finding the general solution means solving the equation to express the dependent variable in terms of the independent variable(s), including an arbitrary constant. The general solution encompasses all possible specific solutions, depending on the initial or boundary conditions given later.It is crucial to distinguish between the general solution and a particular solution:

• **General solution**: Includes arbitrary constants and represents a set of potential solutions.
• **Particular solution**: Comes from applying initial or boundary conditions to the general solution, which then eliminates the arbitrary constants.

In the context of our exercise, we found the general solution to be \(p(x) = -2x^{-8} - 5x + 2x^7 + C\). Here, \(C\) is the arbitrary constant that will be resolved to a specific value once more information, like a point in the solution curve, is provided.

Exploring First-Order Differential Equations

A first-order differential equation involves the first derivative of an unknown function. The equation in our exercise \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^6\) is a prime example. These equations often represent physical processes where the rate of change of a system is dependent on only one independent variable.Understanding first-order differential equations involves the following key points:

• They consist of the first derivative \(p'(x)\), not involving higher derivatives like \(p''(x)\).
• The goal is usually to find the function \(p(x)\) from \(p'(x)\), often via integration.
• They often model real-world scenarios such as velocity rates, population growth, or thermal changes.

First-order differential equations are often simpler to solve than higher-order ones because they require only a single integration step. Thus, understanding these basics provides a strong foundation for tackling more complex differential equations in the future.