MarkFL said:Yes, you are correct. For the left side, consider:
\(\displaystyle \frac{y-2}{y+3}=\frac{y+3-5}{y+3}=1-\frac{5}{y+3}\)
Do the same kind of thing on the right side, and you should be able to integrate now.
find_the_fun said:Is an alternative to \(\displaystyle \frac{y+3-5}{y+3}\) doing polynomial division and seeing y+3 goes into y-2 once with a remainder of 5? I'm not super clear on the thought process of getting \(\displaystyle 1-\frac{5}{y+3}\).
"Solve by separation of variables" is a mathematical technique used to solve differential equations by separating the variables in the equation into different parts and finding the solution for each part separately. This method is often used in physics and engineering to solve problems involving changing rates or quantities.
The first step in solving by separation of variables is to rewrite the equation so that one variable is on one side and the other variable is on the other side. Then, the two sides of the equation can be integrated separately, resulting in two separate equations. These equations can then be combined to find the final solution.
"Solve by separation of variables" is typically used when dealing with differential equations that are separable, meaning that the variables can be separated into different parts. It is commonly used in physics and engineering problems that involve rates of change or quantities that vary over time.
One advantage of using "solve by separation of variables" is that it can simplify complex differential equations and make them easier to solve. It also allows for the use of basic integration techniques, making it a more accessible method for those without advanced mathematical knowledge.
One limitation of using "solve by separation of variables" is that it can only be applied to certain types of differential equations, specifically those that are separable. Additionally, it may not always result in an exact solution and may require additional approximations to find an accurate answer.