For this take home exam you will investigate the double pendulum modeled by the system ODEs (m1 + m2 )l1 2θ1 ′′ + m2l1l2θ2 ′′ + (m1 + m2 )l1gθ1 = 0 m2l2 2θ2 ′′ + m2l1l2θ1 ′′ + m2l2gθ2 = 0 For this problem let m1 = 3, m2 = 1, l1 = l2 = 16 and g = acceleration of gravity. (a) Apply the Laplace transform to the system of equations. The result will be a system of algebraic equations. (b) Solve the system for the two dependent variables. The result will be two rational functions. (c) Use partial fraction decomposition to rewrite the two rational functions (d) Apply the inverse Laplace transform with the initial conditions θ1 (0) = 1, θ1′(0) = 0 θ2 (0) = −1, θ2′(0) = 0