Help solving a second order ODE with repeated roots,

In summary, the conversation discusses a problem with solving a second order ODE with repeated roots. The general solution is found to be y = (Ax + B)e3x, and the goal is to find the values of A and B when given specific values for y and x. The solution is found by plugging in the given values and solving for the unknowns A and B. The conversation ends with the realization that the problem was simpler than originally thought.
  • #1
adam640
19
0
Help solving a second order ODE with repeated roots, urgent!

I have a differential equaition

d2y/dx2 - 6dy/dx + 9y = 0

I have found the general solution to be

y = (Ax + B)e3x

Now I need to find the solutions to A and B so that...

when y = 4, x = 0
when y = 49.e15, x = 5

I literally have no idea where to start with this question and I have this and a few others of a similar format due in a couple of hours, so any help is greatly appreciated. I understand the other methods for ODE's but I do not have any notes for this method and have been unable to find any resources online.

Any help is greatly appreciated!

Thanks,

Adam
 
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  • #2


Plug in y=4, x=0. You get an equation for the unknowns A and B. Then plug in y=49 e15, x=5 (simplify, do not evaluate!) It is an other equation for A, B. Two equations for two unknowns. Solve.

ehild
 
  • #3


Thanks... I didn't think it would be that simple! I was going far too deep into this question.

It's been a long night! haha
 
  • #4


Maths is simpler than you imagine:smile:

ehild
 

1. How do I identify a second order ODE with repeated roots?

In a second order ODE, the highest derivative is squared and there are only two distinct roots. If both roots are the same, it is considered a second order ODE with repeated roots.

2. What is the general form of a second order ODE with repeated roots?

The general form of a second order ODE with repeated roots is: a2y'' + a1y' + a0y = 0, where a2 is the coefficient of the second derivative, a1 is the coefficient of the first derivative, and a0 is the constant term.

3. How do I solve a second order ODE with repeated roots using the method of undetermined coefficients?

To solve a second order ODE with repeated roots using the method of undetermined coefficients, you will need to use the substitution y = v·xn, where n is the multiplicity of the repeated root. This will allow you to find the particular solution for the equation.

4. Can I use the method of variation of parameters to solve a second order ODE with repeated roots?

Yes, the method of variation of parameters can also be used to solve a second order ODE with repeated roots. However, it may be more complicated and time-consuming compared to the method of undetermined coefficients.

5. Are there any special cases or exceptions to consider when solving a second order ODE with repeated roots?

Yes, if the coefficients in the ODE are not constants, but instead depend on the independent variable x, then the method of undetermined coefficients cannot be used. In this case, you will need to use the method of reduction of order or the method of Frobenius to solve the ODE.

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