User:Egm6321.f09.Team06.sada/HW4 make-up

PROBLEM 1.3 b
Solve $$ \frac{d^2y}{dx^2} + 4y = 2{\sec{2x}} $$

Solution

We know that,

$$ y_{Gen} = y_{H} + y_{P} $$

- To Find Homogeneous Solution

$$ y'' + 4y = 0 $$

$$ r^2 +4 = 0 $$

$$ r= +2i, -2i $$

$$ y_{H}= C_{1}{\cos{2x}} + C_{2}{\sin{2x}} $$

- To Find Particular Solution by using Variation of Paramaters

$$ {\cos(2x)}C'_{1} + {\sin(2x)}C'_{2} = 0 $$

$$ -2{\sin(2x)}C'_{1} + 2{\cos(2x)}C'_{2} = 2{\sec(2x)} $$

$$ C'_{1} = -tan(2x) $$ $$ C'_{2} = 1 $$

$$ \int C'_{1} = -\int {tan(2x)} = 2ln{\cos(2x)} = ln{cos^2(2x)} $$

$$ \int C'_{2} = \int 1 = x $$

Thus, we obtain the general solution as

$$ y_{Gen} = y_{H} + y_{P} $$

$$ Y_{Gen} = C_{1}{\cos(2x)} + C_{2}{\sin(2x)} + ln{\cos(2x)} + x{\sin(2x)} $$