For the differential equation (D2 - 9)y = 81x2 Solve the homogeneous equation Th

Question

For the differential equation (D2 - 9)y = 81x2 Solve the homogeneous equation The characteristic polynomial with variable m is The roots of the characteristic polynomial are List the complementary functions (y1. y2) Remember the general solution of the homogeneous equation is yc = ay1 + by 2. yc = Find the particular solution Using the form of the right hand side (see help sheet below), find the particular solution yP = Solve the non-homogeneous equation The equation {D2 - 9)y = 81x2 has general solution y = yc + yP = Now that we have the general solution solve the IVP y(0) = 4 y'(0) = -3 y =

Explanation / Answer

characteristic polynomial = m^2-9=0

roots= +3,-3

Yc= C1e^3x + C2e^(-3x)

Yp=A+Bx^2 (assume)

then substitute in given differential equation we get

D^2(A+Bx^2)- 9(A+Bx^2) = 81x^2

2B-9A-9Bx^2=81x^2

comparing LHS and RHS we get

B= -9 and A=2B/9=-2

so PI= (-2) + (-9x^2)

y=yc+yp= C1e^3x + C2e^(-3x)+ (-2) + (-9x^2)

solving for initial values we get

y(0)=4= C1+C2-2 = C1+C2=6

Y`(0)= -3 = 3C1-3C2

SOLVING WE GET

C1= 2.5

C2= 3.5

SO

y=yc+yp= 2.5e^3x -3.5e^(-3x)+ (-2) + (-9x^2)

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