Solutions of Homogeneous System

In summary, a solution to an ODE can be formed from two existing solutions by taking their linear combination.
  • #1
zohapmkoftid
27
1

Homework Statement



If X0 and X1 are solutions to the homogeneous system of equation AX = 0, show that rX0 + sX1 is also a solution for any scalars r and s.

Thanks for help!

Homework Equations





The Attempt at a Solution

 
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  • #2
zohapmkoftid said:

Homework Statement



If X0 and X1 are solutions to the homogeneous system of equation AX = 0, show that rX0 + sX1 is also a solution for any scalars r and s.

From the given information, what can you say about the values of Ax0 and Ax1?
 
  • #3
Since X0 and X1 are the solutions, therefore
AX0 = 0 and AX1 = 0
Right?
 
  • #4
Right. Now what is A(rx0 + sx1)?

You need to know something about how matrix multiplication works.
 
  • #5
A(rX0 + sX1)
= ArX0 + AsX1
= rAX0 + sAX1
= 0

Thanks for your help
 
  • #6
But how can we prove rX0 + sX1 = 0 from A(rX0 + sX1) = 0
 
  • #7
zohapmkoftid said:
But how can we prove rX0 + sX1 = 0 from A(rX0 + sX1) = 0

This is not what you are supposed to prove.

You are supposed to prove that X = rX0 + sX1 is a solution to the equation AX = 0. You just did that.
 
  • #8
Thanks. I understand now
 
  • #9
zohapmkoftid said:

Homework Statement



If X0 and X1 are solutions to the homogeneous system of equation AX = 0, show that rX0 + sX1 is also a solution for any scalars r and s.

Thanks for help!

Homework Equations





The Attempt at a Solution


You guys forgot to mention that this fact is the socalled super-position principle for ODEs.

Which states if [tex]x_1, x_2[/tex] are solution of the ODE [tex]x' = F(t,x)[/tex]

which states that a solution x can be formed of two existing solutions

such that [tex]x(t) = r \cdot x_1(t) + s \cdot x_2(t)[/tex] is also a solution of the ODE.

This is also called a linear combination for you young wipping snappers out there.

You can read about it here http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx
 
  • #10
I have no idea why you would think anyone "forgot" to say that or why it would need saying. What is given here is a basic property of linear transformations and is used in many applications other that linear differential equations. (There is NO "super-position" principle for general ODEs.)
 

Related to Solutions of Homogeneous System

1. What is a homogeneous system?

A homogeneous system is a set of linear equations in which all the terms have the same degree and are equal to zero. This means that all the unknown variables in the system have the same power.

2. How do you solve a homogeneous system?

To solve a homogeneous system, you can use the method of Gaussian elimination or matrix inversion. You can also use determinants and Cramer's rule to solve for the unknown variables.

3. Can a homogeneous system have a unique solution?

Yes, a homogeneous system can have a unique solution if all the unknown variables are equal to zero. In this case, the system is called trivial.

4. What does it mean if a homogeneous system has infinitely many solutions?

If a homogeneous system has infinitely many solutions, it means that there are more unknown variables than equations, and the system is underdetermined. In this case, there is not enough information to determine a unique solution.

5. How can a homogeneous system be used in real-life applications?

Homogeneous systems are commonly used in fields such as engineering, physics, and economics to model linear relationships between different variables. They can be used to solve optimization problems, analyze data, and make predictions.

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