91Ó°ÊÓ

When solving differential equations, verifying a solution is crucial to ensure correctness. This is done by checking whether a proposed function indeed satisfies the original differential equation. The process typically involves two main steps:

• Take the derivative of the proposed function, if necessary.
• Substitute the function and its derivative back into the differential equation to see if both sides are equal.

In our exercise, we were given a differential equation \( y' = y^2 \) and asked to verify if certain functions are solutions. For verification, once we found the derivatives of the functions, we substituted them alongside the functions into the differential equation. In each case, equality holds, proving that these functions solve the given equation.

Solution verification is a foundational technique in differential equations, ensuring only correct functions are recognized as solutions.

Understanding derivatives is key in working with differential equations. A derivative is the rate at which a function changes at any given point, typically denoted as \( y' \). It provides the necessary link between a function and the solutions to its related differential equations.

For example, consider the function \( y = -\frac{1}{x} \). Its derivative is calculated as \( y' = \frac{1}{x^2} \). Similarly, for functions like \( y = -\frac{1}{x+3} \) and \( y = -\frac{1}{x+C} \), the derivatives are \( y' = \frac{1}{(x+3)^2} \) and \( y' = \frac{1}{(x+C)^2} \), respectively.

Computing derivatives helps simplify the task of checking which functions satisfy given differential equations. Understanding the manipulation of derivatives, particularly within fractions or complex functions, is crucial for this process.

Practicing derivation strengthens your ability to solve not just single problems, but broad applications in calculus and beyond.

Function substitution is a technique used to verify solutions of differential equations. In function substitution, we plug the proposed function and its derivative back into the differential equation. This checks if the left-hand side equals the right-hand side of the equation.

Take, for example, our differential equation \( y' = y^2 \). For each function, such as \( y = -\frac{1}{x} \), after deriving the expression \( y' = \frac{1}{x^2} \), we substitute \( y \) and \( y' \) back into the equation:
\( \frac{1}{x^2} = \left(-\frac{1}{x}\right)^2 = \frac{1}{x^2} \).

Function substitution effectively bridges the calculated derivatives and the algebraic expressions of the functions. By ensuring this equality holds, we confirm the function as a valid solution.

Supporting your solutions with function substitution not only demonstrates accuracy but builds confidence in comprehending the intricate nature of differential equations.