This is almost the general solution. It would be y(t) = Ae2t + 5e8t - 5.goonking said:
No, and it's nowhere near close.goonking said:
A linear differential equation is a mathematical equation that involves an unknown function and its derivatives. It is called linear because the unknown function and its derivatives appear only in the first power, and the equation can be written in the form of a linear combination of the function and its derivatives.
Linear differential equations are important in many areas of science and engineering because they can model many physical phenomena, such as growth, decay, and motion. They also have well-defined solutions, making them useful for predicting future behavior.
The most common method for solving a linear differential equation is by using the technique of separation of variables. This involves isolating the variables on one side of the equation and integrating both sides to find the solution. Other methods include using the method of undetermined coefficients or using Laplace transforms.
Initial conditions refer to the values of the unknown function and its derivatives at a specific point, usually denoted as t=0. These values are necessary to uniquely determine a solution to a differential equation. They can represent physical quantities such as position, velocity, or temperature at a specific time.
Yes, a linear differential equation can have infinitely many solutions. This is because any constant value can be added to a solution without changing its validity. However, for a specific set of initial conditions, there is only one unique solution that satisfies the equation.