91Ó°ÊÓ

Separation of variables is a fundamental technique for solving simple differential equations. Here, we rearrange the equation to isolate each variable on different sides. This makes it easier to integrate and find the solution.

Consider the given differential equation: \(y' = 3t^2 y^2\). In this simple example, separation of variables involves moving all terms involving y to one side and all terms with t to the other.

We achieve this by dividing both sides by \( y^2 \) and multiplying both sides by dt, transforming it into \( \frac{dy}{y^2} = 3t^2 dt \). By separating the variables, we prepare the equation for integration, the next essential step.

Integration plays a crucial role in solving differential equations after the variables are separated. Here, we need to integrate both sides of the equation. For the left side of the equation \( \frac{dy}{y^2} \), the integral is \( -\frac{1}{y} \). And for the right side, the integral of \( 3t^2 dt \) is \( t^3 \) with a constant of integration, C.

The general integral thus becomes \( -\frac{1}{y} = t^3 + C \). Integration helps to find a family of functions that could potentially solve the original differential equation, each defined by different values of the constant C. This process shows how powerful integration is for uncovering the underlying function y(t).

The constant of integration, often represented as C, arises naturally when we perform indefinite integrals. This constant captures the idea that there are infinitely many antiderivatives for a given function, all differing by a constant.

In our solution for the differential equation, once we integrated, we obtained \( -\frac{1}{y} = t^3 + C \). This C represents the different potential solutions that exist because a single-function differentiation results in only one possible derivative. To solve for y, we take the reciprocal and multiply by -1, arriving at \( y = -\frac{1}{t^3 + C} \).

Understanding the concept of the constant of integration is key, as it ensures we've accounted for all possible solutions of the given differential equation.