Solve Differential Equation Using Separation of Variables

Solve the differential equation $\frac{dy}{dx} = xy$ using separation of variables, given the initial condition $y(0) = 5$, and find both the general and particular solutions.

When solving differential equations through the method of separation of variables, the primary objective is to rewrite the differential equation in such a way that allows us to integrate both sides separately with respect to their respective variables. For an equation of the form $\frac{dy}{dx} = g(x)h(y)$, we can manipulate it to $\frac{dy}{h(y)} = g(x) \, dx$, which highlights the separation of the variables. This manipulation leads to two integrals: one in terms of y and the other in terms of x. By integrating both sides, you can solve for y in terms of x, which often requires an integration constant as part of the general solution.

In this problem, you are tasked with solving the differential equation involving the product of the variables x and y. The initial condition provided, $y(0) = 5$, allows you to pinpoint the particular solution among the family of solutions represented by the integration constant. This is crucial, as different initial conditions will yield different particular solutions. Therefore, it's essential to understand the role of initial conditions in differential equations, where they help specify the solution to a differential equation that would otherwise represent a family of curves.

Overall, tackling this kind of problem enhances your ability to work with separable differential equations, which are among the more straightforward types of differential equations encountered in calculus. Mastery of this technique is foundational for progression to more complex types of differential equations, including those that require consideration of linearity or exact solutions.

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