Question: (1 point) We consider the initial value problem x2y - 7xy' + 16y = 0, y(1) = -1, y' (1) = -7 By looking for

(1 point) We consider the initial value problem x2y" - 7xy' + 16y = 0, y(1) = -1, y' (1) = -7 By looking for solutions in the form y = x" in an Euler-Cauchy problem Ax2y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar2 + (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 y1 , y2 : (enter your results as a comma separated list ) (4) Recall that the complementary solution (i.e., the general solution) is y. = 1y1 + C212 . Find the unique solution satisfying y(1) = -1, y' (1) = -7 y =('1 point) We consider the initial value problem xzy\" 5xy' + 931 = 0, y(1) = 3, 3/0) = 6 By looking for solutions in the form y = x' in an Euler-Cauchy problem Axly\" + Bx'y' + Cy = 0, we obtain a auxiliary equation Ar2 + (B A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem nd 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 y1 , 3:2 : (enter your results as a comma separated list ) (4) Recall that the complementary solution (i.e., the general solution) is ye = c1)'1 + czyz. Find the unique solution satisfying y(1) = 3, y'(1) = 6 y: (1 point) We consider the initial value problem x2y" + 15xy' + 49y = 0, y(1) = 2, y' (1) = -15 By looking for solutions in the form y = x" in an Euler-Cauchy problem Ax2y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar2 + (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 y1 , y2 : (enter your results as a comma separated list ) (4) Recall that the complementary solution (i.e., the general solution) is y. = 1y1 + C212 . Find the unique solution satisfying y(1) = 2, y' (1) = -15 y =

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