To show that x and 1+x² are solutions to a differential equation (DE), you need to substitute them into the DE and see if they satisfy the equation. If they do, they are considered solutions to the DE.
A differential equation is an equation that involves an unknown function and its derivatives. It is used to describe relationships between a function and its derivatives in various fields of science and engineering.
X and 1+x² are considered solutions to a DE because when they are substituted into the equation, they satisfy the equation. This means that they can help us find the unknown function that satisfies the DE.
Yes, there can be more than one solution to a DE. In fact, most DEs have an infinite number of solutions. This is because we can add any constant value to a solution and it will still satisfy the DE.
Finding solutions to a DE is important because it helps us understand and predict how a system will behave over time. DEs are used in various fields such as physics, engineering, and economics to model real-world phenomena and make predictions. Solutions to DEs also allow us to solve practical problems and optimize processes.