A differential equation is a mathematical equation that relates a function with one or more of its derivatives. It is used to describe how a system changes over time and is a fundamental tool in many scientific fields, including physics, engineering, and economics.
To solve a differential equation, you need to find the function that satisfies the equation. This can be done through various methods, such as separation of variables, integrating factors, or using a series solution. In this specific case, we can use the method of separation of variables to solve the given differential equation.
The method of separation of variables involves separating the differential equation into two separate equations and then integrating both sides. This method is used when the differential equation can be written in the form of dy/dx = f(x)g(y), where f(x) is a function of x and g(y) is a function of y.
To apply the method of separation of variables to the given equation, we first rearrange the equation to get it in the form of dy/dx = f(x)g(y). In this case, we can divide both sides by e^y and multiply both sides by dx to get dy = [(x^2+1)^(3/2)]/3e^y dx. Then, we can integrate both sides to get the solution for y.
The general solution to the given differential equation is y = ln(x^2 + 1) + C, where C is the constant of integration. This solution can be obtained by integrating both sides of the equation and adding the constant of integration to the right side. It represents all possible solutions to the given differential equation.