Respuesta :
The question is:
Consider the differential equation:
[tex]y''-2y' + 17y = 0 \\ e^xcos 4x, e^x sin 4x\\ $on the interval$ (-\infty, \infty).[/tex]
(1) 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 [tex]W\left(e^x cos 4x, e^x sin 4x \right) \neq 0 \\ $on$ -\infty< x < \infty.[/tex]
(2) Form the general solution.
Answer:
(1) To verify if the given functions form a fundamental set of solutions to the differential equation, we find the Wronskian of the two functions.
The Wronskian of functions [tex]y_1 $and$ y_2[/tex] is given as
[tex]W(y_1, y_2) = \left|\begin{array}{cc}y_1&y_2\\y_1'&y_2'\end{array}\right|\\\\y_1= e^x cos 4x \\y_2 = e^x sin 4x \\y_1' = -4e^x sin 4x \\y_2' = 4e^x cos 4x \\[/tex]
[tex]W\left(e^x cos 4x, e^x sin 4x \right) = \left|\begin{array}{cc}e^x cos 4x &e^x sin 4x \\ \\ -4e^x sin 4x&4e^x cos 4x \end{array}\right|[/tex]
[tex]= 4e^{2x} cos^2 4x + e^{2x} sin^2 4x \\ \\ = 4e^{2x}\left( cos^2 4x + sin^2 4x\right)\\ \\$but $cos^2 4x + sin^2 4x = 1\\ \\W\left(y_1, y_2 \right) = 4e^{2x} \neq 0[/tex]
(2) The general solution may be expressed as a linear combination
[tex]y = C_1e^x cos 4x + C_2e^x sin 4x[/tex]
Where [tex]C_1 $ and $ C_2[/tex]are arbitrary constants.
