Non Homogenous Differential Equation

In summary, a non-homogeneous differential equation is a type of differential equation that contains terms not equal to zero. To solve it, the method of undetermined coefficients is used. The difference between a homogeneous and non-homogeneous differential equation is that the latter also includes terms not involving the dependent variable and its derivatives. A non-homogeneous differential equation can have infinitely many solutions due to the constant of integration in its general solution. Real-life applications of these equations include modeling physical phenomena in various fields such as physics, engineering, and economics.
  • #1
shards5
38
0

Homework Statement


y"' - 9y" +18y' = 30ex
y(0) = 16
y'(0) = 14
y"(0) = 11

Homework Equations



n/a

The Attempt at a Solution


Factor Out
r(r2 - 9r +18)
r = 0; r = 6; r =3
General Equation
y(x) = c0 + c1e3x + c2e6x
y'(x) = 3c1e3x + 6c2e6x
y"(x) = 9c1e3x + 36c2e6x
c1 = (11 - 36c2)/9
y'(x) = 14 = 3c1e3x + 6c2e6x
y'(x) = 14 = ((11 - 36c2)/9)e3x + 6c2e6x
c2 = -31/30
c1 = (11 - 36*(-31/30))/9 = 5.35555556
y(0) = 16 = c0 + c1e3x + c2e6x
16 +31/30 - 5.35555556 = c0 = 11.6777778
Solve For A in Aex
yp = Aex
y'p = Aex
y"p = Aex
y"'p = Aex
Inputting into the original equation we get.
Aex - 9Aex +18Aex = 30ex
Simplifying we get.
10Aex = 30ex
Which gives A = 3 and since 3 is a root of the original equation we add an x to differentiate between the two.
So the final equation SHOULD BE 3xex + 11.6777778 + 5.35555556e3x -31/30e6x but of course its not.
So my question is, what am I doing wrong?
 
Last edited:
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  • #2
Most of your work is fine, but I believe you have made an error in your calculations for c_0, c_1, and c_2. I used matrix methods to solve for these constants and got c_2 = 25/18. I'm fairly confident of this value, but didn't check it.

Also, in your last paragraph you say something that isn't true. You got A = 3, which means that your particular solution is y_p = 3e^x. The fact that you got a value of 3 when you solved for A is irrelevant to anything else in this problem. Your general solution will include 3e^x, not 3xe^x.
 
  • #3
When you say matrix method do you just make a matrix like this?

1 1 1 16
0 3 6 14
0 9 36 11

And then you row reduce?
 
  • #4
shards5 said:
When you say matrix method do you just make a matrix like this?

1 1 1 16
0 3 6 14
0 9 36 11

And then you row reduce?
Yes, and yes.
 

Related to Non Homogenous Differential Equation

1. What is a non-homogeneous differential equation?

A non-homogeneous differential equation is a type of differential equation where the terms involving the dependent variable and its derivatives are not equal to zero. This means that the equation is not in its simplest form, which is known as a homogeneous differential equation.

2. How do you solve a non-homogeneous differential equation?

To solve a non-homogeneous differential equation, we use a method called the method of undetermined coefficients. This involves finding a particular solution to the equation by assuming a general form for the solution and then substituting it into the equation to determine the coefficients. The general solution is then found by adding this particular solution to the general solution of the corresponding homogeneous equation.

3. What is the difference between a homogeneous and non-homogeneous differential equation?

A homogeneous differential equation only contains terms involving the dependent variable and its derivatives, while a non-homogeneous differential equation also includes terms that do not involve the dependent variable and its derivatives. This makes the process of solving non-homogeneous differential equations more complex, as we need to find a particular solution in addition to the general solution.

4. Can a non-homogeneous differential equation have more than one solution?

Yes, a non-homogeneous differential equation can have infinitely many solutions. This is because the general solution of a non-homogeneous differential equation contains a constant of integration, which can take on any value. This means that there are many different solutions that satisfy the given equation.

5. What are some real-life applications of non-homogeneous differential equations?

Non-homogeneous differential equations are used to model various physical phenomena in fields such as physics, engineering, and economics. Some examples include modeling the growth of a population, the spread of disease, and the motion of objects under the influence of external forces. They are also used in signal processing, control theory, and other areas of mathematics.

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