Homogenous constant coefficient linear differential equations

Therefore, the general solution of the equation Y'' + 6Y' +9Y = 0 is Y=C1e-3x+C2xe-3x. In summary, the general solution of the equation is Y=C1e-3x+C2xe-3x.
  • #1
kmikias
73
0
Hello
I am kinda confused when it comes to finding a general solution of equation.
Here is the question.

Y'' + 6Y' +9Y = 0
Solution
Y'' + 6Y' +9Y = 0
I used "x" instead of Lamda...ANYWAY

X^2 + 6X + 9 = 0
FACTOR IT WHICH IS (X+3) (X+3) = 0

X IS EQUAL TO -3.

HERE IS WHERE I GET CONFUSED

WHEN I WRITE THE GENERAL SOLUTION ...DO I SUPPOSE TO WRITE
Y= C1 e-3x + C2 e-3x

or

Y = C1 e-3x

WHICH ONE IS CORRECT?
 
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  • #2
kmikias said:
Hello
I am kinda confused when it comes to finding a general solution of equation.
Here is the question.

Y'' + 6Y' +9Y = 0
Solution
Y'' + 6Y' +9Y = 0
I used "x" instead of Lamda...ANYWAY

X^2 + 6X + 9 = 0
FACTOR IT WHICH IS (X+3) (X+3) = 0

X IS EQUAL TO -3.

HERE IS WHERE I GET CONFUSED

WHEN I WRITE THE GENERAL SOLUTION ...DO I SUPPOSE TO WRITE
Y= C1 e-3x + C2 e-3x

or

Y = C1 e-3x

WHICH ONE IS CORRECT?

Neither is completely correct. A repeated root of r gives solution pair {erx,xerx}.
 

1. What is a homogenous constant coefficient linear differential equation?

A homogenous constant coefficient linear differential equation is an equation that involves the derivatives of an unknown function, where the coefficients of the derivatives are constants and the dependent variable and its derivatives have the same degree. The equation is considered homogenous because all the terms involve the same variable and its derivatives.

2. How do you solve a homogenous constant coefficient linear differential equation?

To solve a homogenous constant coefficient linear differential equation, you can use the method of separation of variables or the method of undetermined coefficients. In both methods, you will need to use the initial conditions to determine the value of the arbitrary constant(s) in the solution.

3. What is the general solution of a homogenous constant coefficient linear differential equation?

The general solution of a homogenous constant coefficient linear differential equation is a family of functions that satisfy the equation. It includes the particular solution, which satisfies the equation and the initial conditions, and the complementary function, which is a linear combination of exponential functions with undetermined coefficients.

4. Can a homogenous constant coefficient linear differential equation have complex solutions?

Yes, a homogenous constant coefficient linear differential equation can have complex solutions. This is because the coefficients of the derivatives are constants, and complex numbers can be used in the coefficients. The real and imaginary parts of the complex solutions will also satisfy the equation.

5. What are the applications of homogenous constant coefficient linear differential equations?

Homogenous constant coefficient linear differential equations have various applications in science and engineering, including modeling physical systems such as circuits, vibrations, and population growth. They are also used in economics to study growth and decay models.

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