91Ó°ÊÓ

An initial condition in differential equations is a value that the solution must satisfy at a particular point. In the given problem, the initial condition is y(0) = 1.
It's important because it allows us to find the specific solution to the differential equation rather than a general one. Here, once we integrate and solve for the constant of integration, the initial condition y(0) = 1 helps us determine this constant accurately.
One essential step is substituting x = 0 and y = 1 into the integrated equation. By doing this, we ensure the solution curve correctly matches the given starting point. This typically narrows it down to a precise solution rather than an infinite number of potential solutions.

Separation of variables is a method used to solve differential equations. The goal is to rearrange the equation so that each variable appears on one side. In the problem given, we start with the equation: \(\frac{dy}{dx} = e^{2x - y}\).
To use separation of variables:

• Rearrange the equation: \(e^y dy = e^{2x} dx\)
• Now, both sides are ready for integration. Notice, all terms involving y are on the left, and x terms are on the right. This allows us to treat the differentials separately and solve the equation more easily.

After achieving this separation, we can integrate each side independently. The method stands out for its simplicity and efficiency in dealing with non-linearities, making it a popular choice for solving many first-order differential equations.

Integration is the process of finding the integral of a function, which is essential in solving differential equations. In this problem, after separation of variables, we need to integrate both sides:
\begin{aligned} &\int e^y dy = \int e^{2x} dx \ &e^y = \frac{1}{2} e^{2x} + C \end{aligned}\ Here, integration transforms the initial differential equation into an algebraic form.
On the left, we integrate \(e^y\) with respect to y and on the right, \(e^{2x}\) with respect to x. After integration, we get an equation that includes a constant of integration (C).
Following integration, the initial condition \(y(0) = 1\) is used to find C. Integration is fundamental for solving differential equations because it simplifies them, enabling us to apply initial conditions or boundary values to find one specific solution.