21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
p'(x) = 16/x⁹ - 5 + 14x⁶

Verified step by step guidance

1. Identify the given differential equation: \(p'(x) = \frac{16}{x^{9}} - 5 + 14x^{6}\). This means the derivative of \(p(x)\) with respect to \(x\) is given by this expression.
2. To find the general solution \(p(x)\), integrate both sides of the equation with respect to \(x\): \(p(x) = \int \left( \frac{16}{x^{9}} - 5 + 14x^{6} \right) \, dx\).
3. Rewrite the integrand to make integration easier: \(\frac{16}{x^{9}} = 16x^{-9}\), so the integral becomes \(\int \left( 16x^{-9} - 5 + 14x^{6} \right) \, dx\).
4. Integrate each term separately using the power rule for integration: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n \neq -1\).
5. After integrating each term, combine the results and add the arbitrary constant of integration \(C\) to express the general solution \(p(x)\).

Key Concepts

General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and contains arbitrary constants representing the family of curves satisfying the equation. For first-order equations, integrating the derivative function yields the general solution plus a constant of integration.

Integration of Functions

Integration is the reverse process of differentiation and is used to find the original function from its derivative. To solve p'(x), integrate each term separately with respect to x, applying power rule integration and adding an arbitrary constant.

Power Rule for Integration

The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C for any real number n ≠ -1. This rule is essential for integrating terms like 16/x⁹ and 14x⁶ by rewriting negative exponents and applying the formula correctly.