Use the method of undetermined coefficients to find the gene

Use the method of undetermined coefficients to find the general solution of the following nonhomogeneous differential equations. When the initial condition is provided, find the particular solution as well.

A.) y 2y + y = 5 cos 2x + 10sin 2x, y(0) = 1, y(0) = 2

B.) y + y = x sin 2x, y(0) = 5/9, y(0) = 2

Solution

A) Char polynomial is m^2-2m+1 = 0 and hence m =1,1

Complementary solution is

y(x) = Ae^x+Bxe^x

As right side is having 5 cos2x +10sin2x the particular solutoin of y would be

y_p = c cos 2x+ D sin 2x

y_p\' = -2c sin 2x+2d cos 2x

y_p\' \' = -4c cos2x -4d sin 2x

Plug in the DE given

-4c cos2x-4d sin 2x-2(-2c sin 2x+2d cos 2x) + c cos 2x+ D sin 2x = 5 cos 2x + 10sin 2x

Or cos2x ( -4c-4d+c) = 5cos 2x and sin 2x (-4d+4c+d) = 10 sin 2x

Equate the coefficients to get

3c+4d = -5 and 4c-3d = 10

Multiply I equation by 4 and ii by 3

12c+16d =-20 and 12c-9d = 30

Subtract 25 d = -50 or d =-2

Substitute in 12c+16d = 20

12c = 52 or c = 13/4

y = Ae^x+Bxe^x+ (13 cos 2x)/4-2 sin 2x

y\' = Ae^x+Bxe^x+Be^x- (13 sin 2x)/2-4cos 2x

y(0) = -1 = A+13/4   Or A = -17/4

y\'(0) = 2 = A+B-4

B = 6-A = 41/4

y(x) =( -17e^x/4)+(41xe^x/4)+ (13 cos 2x)/4-2 sin 2x

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2) Here char equation is m^2+m =0

m =0 or -1

y = Ax+Be^-x

y_p = (Dx²+Ex+f) sin x

yp\'    = (2Dx+E)sinx +( Dx²+Ex+f)cos x

yp\" = 2Dsinx +(2Dx+E) cos x + (2Dx+E)cos x- ( Dx²+Ex+f)(sinx)

Substitute in the given DE