Finding Homogeneous Solutions for a Second Order Differential Equation

In summary, the conversation is about finding the homogenous solutions for the equation y''-4y'+4y=x*e2x. The person has already found one solution using the characteristic function, but is struggling to find the second solution. Another person suggests using a double root of 2 to find the second solution, which would be C2*x*e^(2x). The person appreciates the help.
  • #1
Jalo
120
0

Homework Statement


y''-4y'+4y=x*e2x

I'm trying to find the homogenous solutions of this equation. I know there are two, but I can only find one.

YH=> y''-4y'+4y=0

Homework Equations





The Attempt at a Solution



y''-4y'+4y=0
Using the characteristic function:
a2-4a+4=a <=> (a-2)2=0
Therefore
C1*e2x
is a solution. However, since this is a 2nd order differential equation, I should have two. I can solve the rest of the equation if I can find the 2nd solution. I've tought about it a lot but can't manage to find the answer... All help will be appreciated.
Thanks!
 
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  • #2
If you have a double root 2, then C2*x*e^(2x) is also a homogeneous solution.
 
Last edited:

Related to Finding Homogeneous Solutions for a Second Order Differential Equation

1. What is a "Differential equation root"?

A differential equation root refers to a value or set of values that satisfy a given differential equation when substituted into the equation. In other words, the root is the solution to the equation.

2. How do you find the roots of a differential equation?

Finding the roots of a differential equation involves solving the equation by isolating the dependent variable and setting it equal to 0. This can be done analytically or numerically using various methods such as separation of variables, substitution, or Euler's method.

3. What is the significance of differential equation roots?

The roots of a differential equation represent the points at which the equation is equal to 0 and therefore hold important information about the behavior of the system described by the equation. They can also be used to determine critical points, stability, and other properties of the system.

4. Can a differential equation have multiple roots?

Yes, a differential equation can have multiple roots. This is often the case for higher order equations or equations with complex solutions. It is important to consider all roots when analyzing the behavior of a system.

5. How do you verify if a value is a root of a differential equation?

To verify if a value is a root of a differential equation, simply substitute the value into the equation and see if it satisfies the equation. If the equation holds true, then the value is a root. If not, then the value is not a root.

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