Consider the 3nd order differential operator for y(x): Find the solution space t

Question

Consider the 3nd order differential operator for y(x): Find the solution space to the homogeneous differential equation L(y) = 0. Hint: first find an integer root of the characteristic polynomial, then do long division. Use the method of undetermined coefficients to find a particular solution to L(y) = x. Use the method of undetermined coefficients to find a particular solution to L(y) = e 2x. Use the method of undetermined coefficients to find a particular solution to L(y) = ex. Hint: this will involve a lot of product rule differentiation if you just plug in the correct undetermined coefficients trial solution. An alternate shortcut you might consider would be to factor L so that (D - I) is one of the factors and is the part of L you apply first We do an example like this on Monday November 4. Use your work in a,b,c,d and linearity (superposition) to write down the general solution to

Explanation / Answer

L(y) = y''' - 3y'' + 12y' - 10y

a)

L(y) = 0

y''' - 3y'' + 12y' - 10y = 0

characteristic eq. is

m^3 - 3m^2 + 12m - 10 = 0

by using long division we get

(m-1)*(m^2 - 2m + 10) = 0

=>

m = 1, 1 + 3i, 1-3i

solution is

yc(x) = C1*e^(x) + e^(x)*[ A cos(3x) + B sin(3x) ]

b) L(y) = x

y''' - 3y'' + 12y' - 10y = x

let yp = Ax + B

yp' = A

yp'' = 0

yp'''= 0

yp''' - 3yp'' + 12yp' - 10yp = x

0 - 0 + 12A - 10Ax - 10B = x

compare like terms

A = -1/10

and

12A - 10B = 0

=>

B = -12/100

so,

particular solution is

yp = -x/10 - 12/100

c) L(y) = e^(2x)

let yp = Ae^(2x)

yp' = 2Ae^(2x)

yp'' = 4Ae^(2x)

yp''' = 8Ae^(2x)

yp''' - 3yp'' + 12yp' - 10yp = e^(2x)

8Ae^(2x) - 3*4Ae^(2x) + 12*2Ae^(2x) - 10Ae^(2x) = e^(2x)

=>

10Ae^(2x) = e^(2x)

=>

A = 1/10

so,

particular solution is

yp = (1/10)*e^(2x)

d) L(y) = e^x

let yp = Axe^(x)

yp' = Axe^(x) + Ae^(x)

yp'' = Axe^(x)+2Ae^(x)

yp''' = Axe^(x) + 3Ae^(x)

yp''' - 3yp'' + 12yp' - 10yp = e^(x)

Axe^(x) + 3Ae^(x) - 3Axe^(x) - 6Ae^(x) + 12Axe^(x) + 12Ae^(x) - 10Axe^(x) = e^(x)

=>

    9Ae^(x) = e^(x)

=>

A = 1/9

so,

particular solution is

yp = (1/9)*x*e^(x)

e)

general solution is

y(x) = C1*e^(x) + e^(x)*[ A cos(3x) + B sin(3x) ] + a*( -x/10 - 12/100) + b*((1/10)*e^(2x)) + c*((1/9)*x*e^(x))

or

we can write the genral solution

y(x) = C1*e^(x) + e^(x)*[ A cos(3x) + B sin(3x) ]   - 5x - 6 + 2*e^(2x) - x*e^(x)

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