Can you please provide solutions for question 1 in Section I "HODEs/IVP with constant coefficients?"

I. HODESIIVP with constant eoe'icients. 1. Find a real valued solution to the following initial value problems. Sketch a graph of the solution. a. y\" 6y' + 13y = 0, with y(0) = 1. HO) = 1- b. y\" + 4y' + 4y = 0, with y(0) = 1. y'CO) = -4- c. 6y\" + 7y' + 2y = 0, with 3'03) = 7- Y'Co) = '4- 2. For which values of a (if any) are all solutions of y\" (2a 1)y' + (:05: 1) y = 0 unbounded as t > 00? 3. The characteristic equation of a homogeneous 9'\" order linear Differential Equation with constant coefcients has roots 1' = 0 with multiplicity three, 1' = 2 with multiplicity two, 1' = 3 i 2i with multiplicity two. Write the general solution of the Differential Equation. 4. One solution of the DE 6y\") + 531'\" + 25y\" + 2031' + 4y 2 0 is y : cos(2x). Find the general solution. 11. Reduction of order: 1. The ODE tzy" + Bty' + y = 0 has a solution y1(t) = %for t > 0. Find the general solution. 2. The ODE Zty" Sy' + $31 = O has a solution y1(t) = t3 for t > 0. Find the general solution. III. Undetermined coefficients 1. Find the general solution ofthe ODE y\" + Zy' + y = 93. 2. Solve the WP: 31" y' 2y 2 6x + 69\