Question

12. (a) Suppose that x and y satisfy the following system of differential equations $\dot{x}$ + 2$\dot{y}$ - x = cos?t, $\dot{y}$ - 2x - y = sin ?t. Find the second-order differential equation satisfied by the complex variable z = x + iy. Find the general solution of this differential equation for all real values of ? ? 1. Hence find, for all real values of ? ? 1, the solutions satisfying the conditions x = $\dot{x}$ = y = $\dot{y}$ = 0 at t = 0. (b) For the system of differential equations $\dot{x}$ = 2y, $\dot{y}$ = 3x - y find the stationary solution and sketch the phase diagram. Verify algebraically that the stationary point is a saddle point and find the equation of the stable branch.

          12.
(a) Suppose that x and y satisfy the following system of differential equations
$\dot{x}$ + 2$\dot{y}$ - x = cos?t,
$\dot{y}$ - 2x - y = sin ?t.
Find the second-order differential equation satisfied by the complex variable z = x + iy.
Find the general solution of this differential equation for all real values of ? ? 1. Hence
find, for all real values of ? ? 1, the solutions satisfying the conditions x = $\dot{x}$ = y = $\dot{y}$ = 0
at t = 0.
(b) For the system of differential equations
$\dot{x}$ = 2y, $\dot{y}$ = 3x - y
find the stationary solution and sketch the phase diagram.
Verify algebraically that the stationary point is a saddle point and find the equation of the
stable branch.

12.
(a) Suppose that x and y satisfy the following system of differential equations
ẋ + 2ẏ - x = cos?t,
ẏ - 2x - y = sin ?t.
Find the second-order differential equation satisfied by the complex variable z = x + iy.
Find the general solution of this differential equation for all real values of ? ? 1. Hence
find, for all real values of ? ? 1, the solutions satisfying the conditions x = ẋ = y = ẏ = 0
at t = 0.
(b) For the system of differential equations
ẋ = 2y, ẏ = 3x - y
find the stationary solution and sketch the phase diagram.
Verify algebraically that the stationary point is a saddle point and find the equation of the
stable branch.

Added by Jason C.

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