91Ó°ÊÓ

Separation of variables is a crucial technique for solving differential equations like the one we have here. It involves rearranging the equation so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the other side. For the given problem, we have the differential equation \( \frac{du}{dr} = \frac{1 + \sqrt{r}}{1 + \sqrt{u}} \).
By applying separation of variables, we rewrite it as \((1 + \sqrt{u}) \, du = (1 + \sqrt{r}) \, dr \). This allows us to handle each variable separately, making the problem simpler to solve.

• We treat \( du \) and \( dr \) as differentials and solve the equation by integrating each part.
• The rearranged form makes it easier to manage complex expressions derived from the original equation.

This method is particularly useful for ordinary differential equations, where we work to isolate and integrate each side independently.

Integration is the process of finding the antiderivative of a function. In our example, after separating variables, we need to integrate both sides of the equation: \( \int (1 + \sqrt{u}) \, du = \int (1 + \sqrt{r}) \, dr \).
Integration allows us to "undo" the differentiation present in the equation, moving us closer to a solution. Each side of the separated equation is integrated separately:

• The integral on the left is solved as \( u + \frac{2}{3} u^{3/2} + C_1 \).
• The integral on the right is solved as \( r + \frac{2}{3} r^{3/2} + C_2 \).

Integration requires familiarity with basic antiderivatives and techniques like substitution. Once each integral is solved, we recombine our results. Integration is often an essential step in solving ordinary differential equations, especially those involving products or quotients of functions.

When integrating, it's important to remember the constant of integration. Each indefinite integral introduces a constant because there are infinitely many antiderivatives for a given function. In our problem, the equations \( u + \frac{2}{3} u^{3/2} + C_1 \) and \( r + \frac{2}{3} r^{3/2} + C_2 \) both include their own constants, \( C_1 \) and \( C_2 \), respectively.
After integrating both sides, we combine the results to form the final equation:

• \( u + \frac{2}{3} u^{3/2} = r + \frac{2}{3} r^{3/2} + C \)
• The new constant \( C = C_2 - C_1 \)

This step is crucial because it represents the uncertainty in the initial conditions of the differential equation. Not accounting for integration constants can lead to incomplete solutions. They often help to adjust for the specific context or boundary conditions of the problem.

Ordinary differential equations (ODEs) involve functions of one independent variable and their derivatives. In our exercise, the equation \( \frac{du}{dr} = \frac{1 + \sqrt{r}}{1 + \sqrt{u}} \) is an ODE because it relates \( u \) and \( r \), and their derivatives.
These equations are foundational in mathematics, modeling various natural phenomena, from population dynamics to physical systems. ODEs can be solved by a variety of methods, including separation of variables, as we've used here.

• Understanding how to manipulate and solve ODEs is fundamental for fields like engineering and physics.
• They describe systems where quantities change relative to one another in a predictable manner.

Mastering ODEs allows students to apply mathematical concepts to real-world problems effectively, appreciating the details and nuances involved in dynamic systems.