dextercioby said:
A general solution of a differential equation is a solution that contains all possible solutions to the equation. It includes an arbitrary constant that can be adjusted to fit specific initial conditions.
To find a general solution of a differential equation, the equation is first rearranged so that all terms containing the dependent variable are on one side and all terms containing the independent variable are on the other side. Then, the equation is integrated with respect to the independent variable, resulting in a general solution with an arbitrary constant.
A separable differential equation is one that can be written in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. This allows the equation to be separated into two parts that can be integrated separately to find the general solution.
Finding the general solution allows us to determine the behavior of a system over time and make predictions about its future behavior. It also allows us to solve for specific solutions that satisfy certain initial conditions.
A general solution includes an arbitrary constant, while a particular solution is obtained by assigning specific values to the constant, resulting in a unique solution that satisfies given initial conditions. A general solution can also be used to find multiple particular solutions by varying the value of the constant.
