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Separable differential equations are a specific class of first-order differential equations where you can separate the variables, allowing you to integrate both sides independently. They take the form \( \frac{dy}{dx} = g(y) \cdot h(x) \), where the right-hand side can be expressed as a product of two functions: one in terms of \( y \) and the other in terms of \( x \). This separation is the key to solving these equations.When you encounter a separable differential equation, the main strategy is to:

• Rearrange the equation to separate \( y \) variables on one side and \( x \) variables on the other. This is done by dividing or multiplying both sides as necessary.
• Integrate both sides independently. The left side is integrated with respect to \( y \), and the right side with respect to \( x \).

Let's consider the given problem \( y' = ay^2 \). Here, we recognize it as separable because we can rewrite it as \( \frac{dy}{ay^2} = dx \). This successfully separates \( y \) and \( x \), allowing us to integrate both sides and find a solution.

First-order differential equations involve the first derivative of a function. These equations are prevalent in many real-world applications, ranging from physics to engineering. When you see a differential equation like \( y' = ay^2 \), it tells you about the rate of change of \( y \) with respect to \( x \).Characteristics of first-order differential equations include:

• They contain only the first derivative \( y' \) and not any higher-order derivatives.
• They can often be solved analytically, especially if they fall into a specific category, such as separable, linear, or exact.

For the equation \( y' = ay^2 \), it's a first-order equation because there's only the first derivative of \( y \). After separating variables and integrating, we can solve for \( y \) in terms of \( x \), thus obtaining the general solution. Understanding the structure of first-order equations helps in identifying the appropriate method for finding their solutions.

After obtaining a solution to a differential equation, it's crucial to verify that it satisfies the original equation. Verification ensures the accuracy of your solution and is a vital step in problem-solving.To verify a solution:

• Differentiate the solution to find \( y' \).
• Substitute back into the original differential equation to see if both sides match.

In the exercise, the solution \( y = -\frac{1}{a(x + C)} \) was differentiated to find \( y' \), which yielded \( \frac{1}{a(x + C)^2} \). Substituting back into the original equation \( y' = ay^2 \), confirms that both sides are equal, proving the given function is indeed the correct solution.Remember, verifying solutions helps in reinforcing understanding and ensures no mistakes occurred during integration or algebraic manipulation.