Question

stuck on these few, please help.

3. Systems of equations Convert the following fourth order differential equation into a system of four first order differential equations. Use zi for the independent variables of the first order system.

Answer 1

"Given ODE, d^4 y/d t^4 - 3 d^2 y/dt^2 + 4 y (t) = 0 Let, z_1 = y = > z'_1 = y' = z_2 z_2 = y' = > z'_2 = y"" = z_3 z_3 = y"" = > z'_3 = y'"" = z_4 z_1 = y'"" = > z'_4 = y"""" = + 3 y"" - 4 y = 3 z_3 - 4 z_1 So the associated first order system, z'_1 = z_2 z'_2 = z_3 z'_3 = z_4 z'_4 = 3 z_3 - 4 x_1 Given ODE, d^2 y/dt^2 + 4 y = cos (9/4) t, y (0) = 1, y' (0) = 0. Now the solutions for the homogeneous part, d^2 y/dt + 4 y = 0 is, y_x = c_1 cos 2 t c_2 sin 2 t particular solution, y_p = 1/D^2 + 4 cos (9/4 t) = 1/- (9/4) + 4 cos 9/4 (t) = 16/17 cos (9/9 t) So the general solutions "