Please explain about the principle too.

11. Consider the nonlinear second order equation 02")2 + (8y") y 9y2 = 0. (a) Verify that both y = C1 e ' + C; e" and y = C3 cos(3t) + C4 sin(32') satisfy the equation for any combination of C1, C2, C3, and C4. (b) Verify that neither y = e t + cos(3t) nor y = e" sin(3t) satisfy the equation. Conclude that y = C1 3 ' + C2 3" + C3 cos(3t) + C4 sin(3t) is NOT a general solution of the equation. (c) Do you expect the Principle of Superposition to hold for this equation? Why or why not? ((1) This equation can be factored into a product of two second order homogeneous linear equations. Use this fact to deduce the two solutions in (a). (e) Given the initial conditions y(0) = 1 and y'(0) = 2, nd the particular solution(s). 111. Use the method of Undetermined Coefcients (it works the same way for a fourth order linear equation) to nd the general solution of y\") 16y = t2 41+ 822'. Him: The complementary function is ya = C1 e2! + C2 e'Z' + C3 cos(2t) + C4 sin(2t). 11' v-u-u 1 .I n