A solution to a homogeneous system

In summary, a homogeneous system is a system of linear equations with all constants on the right side equal to zero, resulting in either a unique solution of all zeros or an infinite number of solutions. It differs from a non-homogeneous system which has at least one non-zero constant, resulting in a unique solution with non-zero values or no solution. Solving a homogeneous system is significant in various areas of math and science and can be solved using methods such as Gaussian elimination or Cramer's rule. A homogeneous system can have multiple solutions when it has an infinite number of solutions, resulting in a general solution expressed in terms of free variables.
  • #1
apriljones69
1
0

Homework Statement



I know there is a polynomial that is a solution to the equation:

3/2f(x) - x/2f'(x) - f''(x)=x



Homework Equations





The Attempt at a Solution


I tried many polynomials of degrees 1,2 and 3, but they do not work in my equation
 
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  • #2
What happens when you back-substitute f(x)=x?
 
  • #3
I think you want http://en.wikipedia.org/wiki/Method_of_variation_of_parameters" [Broken] for this problem.
 
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1. What is a homogeneous system?

A homogeneous system is a system of linear equations where all the constants on the right side of the equal sign are equal to zero. In other words, the system has no independent term or constant. This results in a unique solution of all zero values for the variables or an infinite number of solutions.

2. How is a homogeneous system different from a non-homogeneous system?

A non-homogeneous system has at least one non-zero constant on the right side of the equal sign, making it different from a homogeneous system. This results in a unique solution with non-zero values for the variables or no solution at all.

3. What is the significance of solving a homogeneous system?

Solving a homogeneous system is important in many areas of mathematics and science. It helps to determine the existence and uniqueness of solutions, and it is used in solving differential equations, linear algebra, and optimization problems.

4. How do you solve a homogeneous system?

To solve a homogeneous system, you can use various methods such as Gaussian elimination, Cramer's rule, or finding the eigenvalues and eigenvectors of the system's coefficient matrix. These methods help to reduce the system to its simplest form and solve for the variables.

5. Can a homogeneous system have multiple solutions?

Yes, a homogeneous system can have multiple solutions. This happens when the system has an infinite number of solutions due to the absence of independent terms. In this case, a general solution can be expressed in terms of free variables, resulting in an infinite number of possible solutions.

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