Question

Consider the differential equation y '' − 2y ' + 5y = 0; ex cos(2x), ex sin(2x), (−[infinity], [infinity]). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(ex cos(2x), ex sin(2x)

Answer 1

Explanation:

Characteristic equation:

`\lambda^2-2\lambda + 5 =0`

`\lambda = 1\pm2i`

The solution of the ODE is

`y = e^x(C_1 cos(2x) + C_2 sin(2x) )`

Now, very find by taking the first and the second derivative of y.

`y' = e^x(-2 C_1 sin (2x) +2C_2 cos (2x)) +e^x (C_1 cos (2x) + C_2sin(2x))`

`=-2e^xC_1 sin(2x) +2e^xC_2 cos (2x)+C_1e^xcos(2x) + C_2e^x sin(2x)`

`y`

`=-4C_1e^xsin(2x) -3C_1e^x cos(2x) +4C_2e^xcos(2x) -3C_2e^xsin(2x)`

Now, put all in y" -2y'+5y and consider if it = 0 or not.