Method of undetermined coefficients

[danield]

Homework Statement

http://edugen.wiley.com/edugen/courses/crs3098/rc/boyce3346c07/math/math1074.gif

solve using the method of undetermined coefficients

Homework Equations

v(t)=a*t*e^-t + be^-t + ct + d

The Attempt at a Solution

what i could do is that i replaced v(t) for x in the equation and got the following
Aa=-a
Ab=a-b-(1,0)
Ac=-(0,1)
Ad=c

now I am not sure how to continue from here i was told that vector a = (alpha, alpha)

to plug this in into the Ab equation so it ends up like this

(A+I)b=(alpha -1, alpha)

but that is giving me that 1 = 0 and I am stuck there
any help is appreciated

 

[mighty2000]

Thank you for your post. I am a scientist and I would be happy to assist you with solving this problem using the method of undetermined coefficients.

Firstly, let's review the problem and the equations you have provided. The problem is asking you to solve the given differential equation using the method of undetermined coefficients. The equation is in the form of v(t) = a*t*e^-t + be^-t + ct + d, and you have correctly replaced v(t) with x to make it x = a*t*e^-t + be^-t + ct + d.

Next, you have started to calculate the coefficients by equating the coefficients of the terms with the same powers. However, there are a few errors in your calculations. Firstly, the coefficient of e^-t term should be (-a+b), not (a-b). Secondly, the coefficient of the constant term should be (-c), not (c). Lastly, the coefficient of the t term should be (c), not (-c).

Now, let's continue with the calculation of the coefficients. We can rewrite the equations as follows:
Aa = -a
Ab = -a+b
Ac = c
Ad = -c

From the first equation, we can see that a = 0. Then, substituting this value into the second equation, we get b = 0. Now, we can substitute both values of a and b into the third equation, we get c = 0. Finally, substituting all the values of a, b, and c into the last equation, we get d = 0.

Therefore, the solution to the differential equation is x = 0. This means that the function v(t) is equal to 0, which is a constant value. This is a trivial solution and it is not very useful. To get a non-trivial solution, we need to use the method of variation of parameters.

I hope this helps you to solve the problem. If you have any further questions or need clarification, please do not hesitate to ask. Best of luck with your calculations!