Do Question 1 Find a general solution of y\" - 4y = e^2x. Given the differential

Question

Do Question 1 Find a general solution of y" - 4y = e^2x. Given the differential equation y" + 9y = 7x cos (3x), set up the appropriate form of the general solution (complementary function y_r + particular solution y_p) but do not determine the value of the coefficients. Given the differential equation y" - 16y = 3x^1 e^1x, set up the appropriate form of the general solution (complementary function y_r + particular solution y_p) but do not determine the values of the coefficients. Find a general solution of the differential equation y^(x) = 4y' = 3x - 1 Solve the initial value problem y" + 5y' + 6y = e^x, y(0) = 0. y'(0) = 2 Use the method of variation of parameters of parameters to find a particular solution of y" + y = 2 sec (x) Find a particular solution of g^(3) + 2y" - 3y' = 3 + 4e^x + 3e^3x Let W be the set of all vectors (x_1, x_2, x_3, x_4) in R^1 such that x_1 x_2 = x_1 x_1. Is W a subspace of R^4? (Justify your answer) Given the following system of linear equations, find two solution vectors u and v such that the solution space is the set of all linear combination of the form (u + v) x_1 - 4x_2 - 3x_3 - 7x_4 = 0 2x_1 - x_2 + x_3 + 7x_1 = 0 x_1 + 2x_2 + 3x_3 + 11x_4 = 0 Determine whether the given vectors are linearly dependent or independent. u = (-2, 3, -5), u = (-4,7,10), w = (5, 6, 8).

Explanation / Answer

1) we have given y''-4y=e2x

first we solve the homogeneous equation

y''-4y=0

this characteristic equation is r^2-4=0

so roots are r=2,-2

complementary solution is yc=c1e^2x+c2e^-2x

now we need to find particular solution for given equation

y''-4y=e2x

using the method of undetermined coefficients, guess that

yp=Axe2x

y'p=2Axe2x+Ae2x

y''p=4Axe2x+2Ae2x+2Ae2x

substitute yp,y'p and y''p into given equation

y''-4y=e2x

4Axe2x+2Ae2x+2Ae2x-4Axe2x=e2x

4Ae2x=e2x

4A=1 implies A=1/4

plug the A=1/4 inot yp

yp=(1/4)xe2x

general solution for given equation is y=yc+yp

y=c1e^(2x)+c2e^(-2x)+(1/4)xe2x

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