Need the answer for PART 3 for all these questions! We consider the initial valu

Question

Need the answer for PART 3 for all these questions!

We consider the initial value problem x^2y" + xy' + 4y = 0, y(1) = -2, y'(1) = -4 By looking for solutions in the form y = x^r in an Euler-Cauchy problem Ax^2y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar^2 +(B - A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. For this problem find the auxiliary equation: = 0 Find the roots of the auxiliary equation: (enter your results as a comma separated list) Find a fundamental set of solutions y_1, y_2: (enter your results as a comma separated list) Recall that the complementary solution (i.e., the general solution) is y_c = c_1y_1 + c_2y_2. Find the unique solution satisfying y(1) = -2, y'(1) = -4 y = -2(sin(2 ln x) + cos(2 ln x)) We consider the initial value problem 9x^2y" - 9xy' + 13y = 0, y(1) = -3, y'(1) = 1 By looking for solutions in the form y = x' in an Euler-Cauchy problem Ax^2y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar^2 + (B - A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list) (3) Find a fundamental set of solutions y_1, y_2: (enter your results as a comma separated list) (4) Recall that the complementary solution (i.e., the general solution) is y_c = c_1y_1 + c_2y_2. Find the unique solution satisfying y(1) = -3, y'(1) = 1 y = We consider the initial value problem x^2y" + 3xy' + 5y = 0, y(1) = -2, y'(1) = 4 By looking for solutions in the form y = x' in an Euler-Cauchy problem Ax^2y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar^2 + (B - A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list) (3) Find a fundamental set of solutions y_1, y_2: (enter your results as a comma separated list) (4) Recall that the complementary solution (i.e., the general solution) is y_c = c_1y_1 + c_2y_2. Find the unique solution satisfying y(1) = -2, y'(1) = 4 y =

Explanation / Answer

1. y=Acos(2logx)+Bcos(2logx)

2. y= x(Acos((2/3)logx)+Bsin((2/3)logx))

3. y=(1/x)((Acos((2)logx)+Bsin((2)logx))

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