Problem with general solution to a complex roots diffy Q

In summary, the conversation discusses finding the real-valued general solution of a differential equation with undetermined constants. The solution involves using the characteristic equation and the quadratic formula, resulting in an equation with e^(-.5x) and trigonometric functions.
  • #1
batmankiller
21
0

Homework Statement




Question: I'm confused about how to app
Find the real-valued general solution of the differential equation

y''+1y'+1y=0
where primes indicate differentiation with respect to x. (Use the parameters a, b, etc., for undetermined constants in your solution.)

The Attempt at a Solution


My attempt:
Use characteristic equation:
r^2+r+1=0
I used the quadratic formula and got r=(-1 +-sqrt(-3))/2
So we get a=-.5 b-sqrt(3)/2

Following e^(ax)(c1cos(bx)+c2sin(bx):

we get e^(-.5x)(a*cos([tex]\sqrt{3}[/tex]*x/2)+bsin([tex]\sqrt{3}[/tex]*x/2))

Anyone see any flaw i nlogic of algebra or math? I can't seem to get a correct answer
 
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  • #2
Your work is correct for the equation given. Is there supposed to be a non-homogeneous term on the right side?
 
  • #3
Nope, it's given as a homogeneous equation (set equal to 0). Unless you're saying there's something I'm missing?
The full equation is y=e^(-.5x)(a*cos(*x/2)+bsin(*x/2))
 
  • #4
batmankiller said:
Nope, it's given as a homogeneous equation (set equal to 0). Unless you're saying there's something I'm missing?
The full equation is y=e^(-.5x)(a*cos(*x/2)+bsin(*x/2))

What led you to believe it was wrong? (Don't forget the sqrt(3)'s).
 
  • #5
Eh it's online homework and it says "answer is incorrect" -.-"
I entered:
e^(-.5x)(a*cos(sqrt(3)*x/2)+bsin(sqrt(3)*x/2)) and my previewed answer was in the correct format too
 
  • #6
Ok so apparently it accepted e^(-x/2) but not e^(-.5x). Thank you so much for your help!
 

FAQ: Problem with general solution to a complex roots diffy Q

What is a complex roots diffy Q?

A complex roots diffy Q is a differential equation that has complex roots as solutions. These roots involve imaginary numbers, which are numbers that involve the square root of -1, known as "i". They can also involve real numbers.

Why is there a problem with the general solution to a complex roots diffy Q?

The general solution to a complex roots diffy Q may not accurately represent the behavior of the system over time. This is because the general solution only takes into account the real solutions, while ignoring the complex solutions. This can lead to inaccuracies and errors in the solution.

How do complex roots affect the behavior of a differential equation?

Complex roots can cause oscillations and rapid changes in the behavior of a differential equation. This is because complex roots can involve imaginary numbers, which can lead to exponential growth or decay in the solution. As a result, the system may behave differently than predicted by the general solution.

Can the general solution be modified to account for complex roots?

Yes, the general solution can be modified to include complex roots by using complex numbers in the solution. This modified solution is known as the complex general solution and can accurately represent the behavior of the system over time, including any oscillations or rapid changes caused by complex roots.

How can the problem with the general solution to a complex roots diffy Q be resolved?

The problem with the general solution can be resolved by using the complex general solution or by approximating the solution using numerical methods. These methods take into account both real and complex solutions, providing a more accurate representation of the behavior of the system. It is important to understand the nature of the differential equation and its solutions in order to appropriately address the problem.

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