TranscendArcu said:So I know [itex]e^{sin(x) + C} = e^{sin(x)}e^C[/itex]. But since [itex]e^C[/itex] is just a constant, can't I just compress it into the constant C?
TranscendArcu said:so I should write,
[itex]\frac{1}{1-y} = e^{sin(x)}C[/itex]
[itex]1-y = \frac{1}{e^{sin(x)}C}[/itex], and C is totally arbitrary so write,
[itex]y = \frac{C}{e^{sin(x)}} + 1[/itex]
The Separation of Variables method is a technique used to solve ordinary differential equations (ODEs) by separating the variables into two or more equations, each with only one variable. This allows for simpler integration and solution of the equations.
To solve a Differential Equation by Separation of Variables, you first need to rewrite the equation in the form dy/dx = f(x)g(y). Then, you can integrate both sides with respect to their respective variables, x and y. This will result in a general solution, which can be further simplified by applying initial conditions.
Separation of Variables can be applied to first-order and second-order ordinary differential equations, as well as some partial differential equations. However, it is not applicable to all types of differential equations, such as those with non-separable terms or those that require numerical methods.
The Separation of Variables method is a straightforward and systematic approach to solving differential equations. It allows for the use of known integration techniques, making the process more manageable. Additionally, it can be applied to a wide range of equations, making it a versatile method.
While Separation of Variables is a useful method for solving differential equations, it is not always applicable. It can only be used for equations with separable variables, and some equations may require additional techniques or numerical methods. Additionally, it may not always produce an explicit solution and may require further manipulation or approximation to obtain a final solution.