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The method of variable separation is a foundational technique in solving differential equations. It involves rearranging the differential equation so that each variable appears on a different side of the equation. This method enables us to integrate each side with respect to its own variable. In our example, the equation starts as: \[3 y^{2} \frac{dy}{dt}=-\text{sin}(t)\]To achieve separation, we multiply both sides by \(dt\) and divide by \(3 y^{2}\). This results in:\[\frac{1}{3 y^{2}} dy = -\text{sin}(t) dt\]Here, we successfully separated variables: \(y\) terms on the left, \(t\) terms on the right. This separation allows for the next step: integration of each side separately. Ensuring variables are correctly separated is crucial for the solving process to work.

Initial conditions are given values that a solution must satisfy, often used to determine unknown constants after integration. In our example, the initial condition is:\[ y\left(\frac{\pi}{2}\right)=1 \]This means when \(t = \frac{\pi}{2}\), \(y = 1\). Once we separate the variables and integrate both sides, we get:\[-\frac{1}{3y} = \text{cos}(t) + C \]Using the initial condition, substitute \(t = \frac{\pi}{2}\) and \(y = 1\):\[ -\frac{1}{3 \cdot 1} = \text{cos}\left(\frac{\pi}{2}\right) + C \]Since \(\text{cos}\left(\frac{\pi}{2}\right) = 0\), we simplify:\[ -\frac{1}{3} = 0 + C \]Thus, \(C = -\frac{1}{3}\). Initial conditions personalize the general solution to fit specific scenarios, making them essential in differential equations.

Integration is the mathematical process of finding the integral of a function, often reversing differentiation. In our context, integrating both sides of a separated differential equation provides the antiderivatives essential to solving the equation. From our separated variables:\[\int \frac{1}{3 y^{2}} dy = \int -\text{sin}(t) dt\]These integrals are computed as follows:For the left side:\[\int \frac{1}{3 y^{2}} dy = \int \frac{1}{3} y^{-2} dy = \frac{1}{3} \, \left( \frac{-1}{y} \right) = -\frac{1}{3y}\]For the right side:\[\int -\text{sin}(t) dt = \text{cos}(t) + C\]Combining these results, we obtain:\[-\frac{1}{3y} = \text{cos}(t) + C\]This step-by-step integration crucially leads us from a differential equation to an equation we can solve for the unknown function \(y(t)\). When tackling differential equations, precise integration and careful manipulation of constants are key to finding accurate solutions.