How to Solve a Differential Equation Using the Superposition Approach?

[ultra100]

Solve this equation by superposition approach (undetermined coefficients); pg. 154 in Zill, according to the formula Y = Yc + Yp

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y''' -2y'' -4y' +8y = 6xe^(2x)

I got:
Yc = C1 e^(-2x) + C2 e^(2x) + C3 xe^(2x)
for the complementary function Yc...


I'm having trouble finding the particular solutions Yp

Any help is greatly appreciated

 

[AKG]

Your book should give you a standard way to find a particular solution when the right sides is of the form P(x)e^kx where P(x) is a polynomial and k is some real number. I believe the particular solution should take the form Q(x)e^kx where Q and P have the same degree. Since in your problem, P(x) = 6x is a degree 1 polynomial, let Q(x) = ax + b be an arbitrary degree 1 polynomial. Plug in y = (ax+b)e^2x into your differential equation, and solve for a and b.

 

[ultra100]

since 6xe^(2x) matches the general term C3xe^(2x), the 6x has to be raised in power to 6x^2; in that case the equation is (Ax^2 + Bx +C)e^(2x)... but when I try to take all 3 derivatives of this function and then substituting them back into the y''' - 2y'' - 4y' + 8y equation, it doesn't match up...

i get 8Ae^(2x) = 6x^2e^(2x), but the A constant needs an X^(2) in order to work, otherwise you can't solve for it

any way I can resolve that?

 

[AKG]

Show all your work, otherwise how can we tell where you're messing up.

 

[ultra100]

The solution formula is Y = Yc + Yp

Yc = (C1 e^-2x) + (C2 e^2x) + (C3 xe^-2x) <--- since this C3 matches 6xe^2x, the rule is that it has to increased in power and it becomes 6x^2e^2x

the formula to find Yp when the polynomial in front of e^2x is of 2nd power is
(Ax^2 + Bx + C) e^2x

y = (Ax^2 + Bx + C) e^2x
y' = 2Axe^2x + 2Ax^2e^2x + Be^2x +2Bxe^2x + 2Ce^2x
y'' = 2Ae^2x + 8Axe^2x + 4Ax^2e^2x + 4Be^2x + 4Bxe^2x + 4Ce^2x
y''' = 12Ae^2x + 24Axe^2x + 8Ax^2e^2x + 12Be^2x + 8Bxe^2x + 8Ce^2x

So i take that new form y, and it's three derivatives (y', y'', y''') and plug it back into the original y''' - 2y'' - 4y' + 8y and set it equal to 6x^2e^2x

everything cancels and I'm left with 8Ae^2x = 6x^2e^2x

I can't solve for A however, because it doesn't have equal powers with 6x^2
if it were 8Ax^2e^2x then I could cancel the X's and solve for the A constant

 

[AKG]

Okay, I did a little reading and I think this is the approach: if the right hand side is P(x)e^kx, then your particular solution takes the form x^sQ(x)e^kx where deg(Q) = deg(P), and s is the multiplicity of k as a root of the characteristic polynomial of the differential equation. In your case, the characteristic polynomial is:

z³ - 2z² - 4z + 8
= (z - 2)²(z + 2)

k = 2, and P(x) = 6x. So set Q(x) = (ax+b), and set s = 2, since k (which is 2) is a double root of your characteristic polynomial.

y_p = x²(ax+b)e^2x = (ax³ + bx²)e^2x.

Plug this into your diff. eq. and solve for a and b.

 

[ultra100]

o wow! that works out... thank's a lot for all your help

 

[AKG]

I always go back to this site when I get rusty with basic differential equation theory.