What's the Mistake in Transforming the Differential Equation Bases?

  • Thread starter kasse
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In summary, you attempted to solve for the two complex conjugates of the characteristic equation, which is two complex numbers that have the property that a^2-4b=-4. However, you ended up finding a different equation that has the same roots, but is different in terms of its form.
  • #1
kasse
384
1

Homework Statement



Find a differential eq. y''+ay'+by=0 with this basis: exp(-1-i)x , exp(-1+i)x

2. The attempt at a solution

I see from this that the solutions of the characteristic eq. is two complex conjugates, and that a^2-4b=-4.

Substitution gives me the eq. -2i-a+ai+b=0, and combining these two gives me

a=(1/2) v (1/2)-i

Then I calcutale b when a=(1/2)

b=(17/16)

So that the eq. is

y''+0.5y'+(17/16)y=0

Looks quite nice, but when I transform the bases (exp(-x)*sinx and exp(-x)*cosx) and substitute, it proves not to be correct.

What's my mistake?
 
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  • #2
kasse said:

Homework Statement



Find a differential eq. y''+ay'+by=0 with this basis: exp(-1-i)x , exp(-1+i)x

2. The attempt at a solution

I see from this that the solutions of the characteristic eq. is two complex conjugates, and that a^2-4b=-4.

Can you explain how you got the above equation?
 
  • #3
learningphysics said:
Can you explain how you got the above equation?

The roots of the characteristic eq. are 0.5(-a (+/-) sqrt(a^2-4c))

Since the two solutions are complex numbers, a^2-4c must be negative, more precisely -4, since the imaginary part of the complex solutions is (+/-) 1.
 
  • #4
kasse said:
The roots of the characteristic eq. are 0.5(-a (+/-) sqrt(a^2-4c))

Since the two solutions are complex numbers, a^2-4c must be negative, more precisely -4, since the imaginary part of the complex solutions is (+/-) 1.

I see... you must have made a mistake when you solved the two equations... I'd just take the sum and product of the roots... then you can write out the equation:

s^2 - qs + p... where q is the sum of the roots and p is the product...

then you get a = -q and b = p for the differential equation

or you can just multiply out:

(s - (-1 - i))(s - (-1 + i))
 
  • #5
Ah, so I use the superposition principle and substitute the sum of Y1 and Y2 into the original eq?
 
  • #6
No, I'm not using the superposition principle here. Since you know that the basis is exp(-1-i)x , exp(-1+i)x... you immediately know that the characteristic equation has roots (-1-i) and (-1+i).

or another way to think about it... you know that the characterstic equation is:
s^2 + as + b

This means that any function of the form e^sx that satisfies the equation must have the property that s^2 + as + b =0... ie s=(-1-i) must satisfy this equation, and s = (-1+i) must satisfy this equation... ie these are the two roots of the equation... Once you know the roots of a degree-2 equation r1 and r2... then you immediately know the form of the equation... s^2 - (r1+r2)s + r1r2... which you can also get by (s-r1)(s-r2)

You can also plug in -1-i into the equation and -1+i into the equation, then get 2 equations in 2 unknowns (a and b) and solve... but this is a little more roundabout way to do the problem I think...
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes the rate of change of a quantity over time or space.

2. What is the process of transforming a differential equation?

The process of transforming a differential equation involves manipulating the equation to change its form, while still maintaining its solution. This can be done through various methods such as substitution, integration, or linearization.

3. What is the purpose of transforming a differential equation?

Transforming a differential equation can make it easier to solve or analyze. It can also help in understanding the behavior of the system described by the equation and making predictions about its future state.

4. What are the common mistakes in transforming a differential equation?

Some common mistakes in transforming a differential equation include errors in algebraic manipulation, incorrect application of transformation techniques, and overlooking initial or boundary conditions.

5. How can one identify the mistake in transforming a differential equation?

To identify a mistake in transforming a differential equation, one can compare the transformed equation with the original equation and check for any discrepancies. It is also helpful to double check the steps taken in the transformation process and ensure that all conditions are taken into account.

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