Dick said:v=y^3 is certainly a good substitution to use, and e^(-x)*(x+c) is certainly the correct answer. What I don't understand is how you used Bernoulli's equation to get a wrong answer. Can you explain?
A differential equation is a mathematical equation that relates a function with one or more of its derivatives. It is used to model and describe the relationship between changing quantities in various scientific fields, such as physics, engineering, and economics.
Daniel Bernoulli was a Swiss mathematician who made significant contributions to the development of differential equations. He introduced the concept of "Bernoulli's equation," which is a type of differential equation used to model the flow of fluids. He also developed the Bernoulli principle, which explains the relationship between pressure and speed in a moving fluid.
An ordinary differential equation involves only one independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are used to describe systems that change over time, while partial differential equations are used to describe systems that change over both time and space.
A Bernoulli differential equation is solved by using a substitution technique. This involves substituting the original dependent variable with a new variable, which transforms the equation into a linear differential equation. Then, the linear equation can be solved using standard techniques, and the solution is then substituted back to find the solution to the original Bernoulli equation.
Bernoulli differential equations have many real-life applications, such as predicting the airflow around an airplane wing, determining the optimal shape of a sail for a boat, and modeling the spread of infectious diseases. They are also used in economics to model supply and demand, in finance to model stock prices, and in chemistry to model chemical reactions.