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A separable differential equation is a type of differential equation in which the variables can be separated on different sides of the equation. This means that you can write the equation so that all instances of one variable are on one side and all instances of the other variable are on the other side. This form makes it easier to integrate and solve. For instance, in the differential equation \(\frac{dy}{dt} = t^{1/2} y^{2}\), we can rewrite it in such a way that allows us to separate the variables and proceed with solving the equation. This leads us to the next important steps of integration and solving for constants.

Once we have separated the variables, the next step is to integrate each side of the equation. Integration is a fundamental technique in calculus used to compute the antiderivative of a function. In the given example, for the differential equation \(\frac{dy}{dt} = t^{1/2} y^{2}\), after separation, we have: \(\frac{1}{y^{2}} dy = t^{1/2} dt\). Here, we integrate both sides:

• For the left side: \( \int \frac{1}{y^{2}} dy = -\frac{1}{y} \)
• For the right side: \(\frac{ \int t^{1/2} dt = \frac{2}{3} t^{3/2} \)

Through these integration steps, we transform the original equation into a more solvable form.

When we integrate each side of the differential equation, we must include a constant of integration on the right-hand side. This constant represents the unknown part of the general solution of the differential equation. For example, after integrating, we obtain: \( - \frac{1}{y} = \frac{2}{3} t^{3/2} + C \). Here, \(C\) is the constant of integration. The value of this constant is usually determined using initial conditions or boundary conditions provided in the problem.

The technique of separation of variables is a method used to solve separable differential equations. By isolating each variable on different sides of the equation, we can integrate both sides independently. For the given differential equation \(\frac{dy}{dt} = t^{1/2} y^{2}\), our goal is to rewrite it as: \( \frac{1}{y^{2}} dy = t^{1/2} dt \). This separation allows us to solve the equation step by step. After integrating and solving for the function \(y\), we can express the solution typically in the form \(y = f(t)\). Thus, through separation of variables, integration, and management of the constants of integration, we arrive at a complete solution to the differential equation.