THis essay focuses on dierential equation and the initial conditions.Solve the initial-value problem consisting of this dierential equation
1. Consider the dierential equation d 3y dx3 + 2x dy dx 4y = x 4 e x . (a) Is this a linear dierential equation? (2 pts.) (b) What is the order of the dierential equation? (2 pts.) 2. Verify that y = xe4x is a solution to the dierential equation y 00 8y 0 + 16y = 0. (4 pts.) 3. The two-parameter family of functions y = c1x 6 + c2x 2 solves the dierential equation x 2 y 00 7xy0 + 12y = 0. Solve the initial-value problem consisting of this dierential equation and the initial conditions y(1) = 4 and y 0 (1) = 6. (4 pts.) 4. Solve the dierential equation. (4 pts. each) (a) dy dx = x 3 y.
A parameter is a quantity that influences the output or behavior of a mathematical object but is viewed as being held constant. Parameters are closely related to variables, and the difference is sometimes just a matter of perspective. Variables are viewed as changing while parameters typically either don’t change or change more slowly. In some contexts, one can imagine performing multiple experiments, where the variables are changing through each experiment, but the parameters are held fixed during each experiment and only change between experiments.
One place parameters appear is within functions. For example, a function might a generic quadratic function as
Here, the variableis regarded as the input to the function. The symbols , , and are parameters that determine the behavior of the function . For each value of the parameters, we get a different function. The influence of parameters on a function