91Ó°ÊÓ

When we talk about the general solution of a differential equation, we're looking for a formula that describes a family of functions accounting for a wide range of solutions to the differential equation. Each specific solution is derived by adjusting the constant of integration from the general formula.
In the case of the exercise above, we arrived at the general solution after performing integration. The relation established was \[ y = (6x^2 + C_1)^{1/3} \]This expression represents all possible solutions of the original differential equation. As you can see, the

• Variable \( C_1 \) is the constant of integration, highlighting that there are infinite functions that satisfy the equation depending on the value of \( C_1 \)
• For each specific problem or initial condition provided, adjusting \( C_1 \) will yield a specific solution.

The beauty of the general solution lies in its flexibility to adapt to various scenarios when additional conditions are specified.

Separable Equations are a category of differential equations where variables can be neatly segregated, allowing for simpler solutions. These equations can be rewritten so that all instances of one variable (e.g., \( y \)) and its differential (\( dy \)) are on one side of the equation, and the other variable (e.g., \( x \)) and its differential (\( dx \)) are on the opposite side.
In this exercise, we took the given equation \( y^2 y' = 4x \)and successfully rewrote it as \( y^2 \, dy = 4x \, dx \).This manipulation is key because:

• Once separated, you can integrate each side independently with respect to its variable.
• This method simplifies the solution process by transforming a differential equation into two simple integrals.

The process of separating the variables enabled us to eventually find our general solution, thereby affirming that the given equation was, indeed, separable.

Integration is a fundamental concept in calculus that involves finding the antiderivative or the area under a curve. It's particularly useful for solving differential equations.

• In the context of separable equations, once we have isolated variables with their respective differentials, integration becomes the next step.
• For the given exercise, we performed the integral operation on both sides: \( \int y^2 \, dy \) and \( \int 4x \, dx \).Each integral was solved independently.

The outcomes of these integrals directly contributed to constructing the general solution. Specifically:

• The left side, integrated to \( \frac{y^3}{3} \), represents the antiderivative of \( y^2 \).
• The right side, integrated to \( 2x^2 + C \), reflects the antiderivative of \( 4x \), with \( C \) being the constant of integration.

This integration step is pivotal as it transforms the problem from one of differential calculus to simple algebraic manipulation to express the general solution.