Finding a linear differential equation

• goonking
In summary, the conversation discusses finding the general solution to a nonhomogeneous differential equation. The general solution is the sum of the complementary solution and the particular solution. The complementary solution is also referred to as the null solution. The particular solution is a solution to the nonhomogeneous problem. The correct general solution is y(t) = Ae2t + 5e8t - 5. The next step is to find the nonhomogeneous differential equation that has y(t) as its solution. It is recommended to refer to the textbook for examples.
goonking

The Attempt at a Solution

So the general solution is the sum of the null solution (Yn) and particular solution (Yp)

I believe I just need to write:

y = e2t + 5e8t - 5 + C

and then find the derivative of both sides

y' = 2e2t + 40e8t

is this correct?

goonking said:

Homework Statement

View attachment 101985

The Attempt at a Solution

So the general solution is the sum of the null solution (Yn) and particular solution (Yp)

I believe I just need to write:

y = e2t + 5e8t - 5 + C
This is almost the general solution. It would be y(t) = Ae2t + 5e8t - 5.
The first part is the complementary solution (the solution to the homogeneous diff. equation) -- what you're calling the null solution. The second part is the particular solution, a solution to the nonhomogeneous problem.
goonking said:
and then find the derivative of both sides

y' = 2e2t + 40e8t

is this correct?
No, and it's nowhere near close.
What you're supposed to do is find the nonhomogeneous differential equation that has y(t) as its solution.Your textbook should have some examples that are similar to this problem. That should be your first resource.

goonking

1. What is a linear differential equation?

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.

2. Why is it important to find linear differential equations?

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.

3. How do you solve a linear differential equation?

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.

4. What are initial conditions in a linear differential equation?

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.

5. Can a linear differential equation have multiple solutions?

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.

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