Question

Verify that the fucntion y=ae²ˣ+be⁻ˣ is a solution of the differential equation d²y/dx² - dy/dx - 2y=0.

Answer 1

Answer:

The solution is

$y=A\:e^{2x}+B\:e^{-x}$

Step-by-step explanation:

Verify that the fucntion y=ae²ˣ+be⁻ˣ is a solution of the differential equation d²y/dx² - dy/dx - 2y=0.

$\frac{d^2y}{dx^2}-\frac{dy}{dx}-2y=0$

Characteristic equation is

$p^2-p-2=0$

$(p-2)(p+1)=0$

$\implies\:p=2,-1$

$\therefore\text{ The roots are real and unequal}$

The complementary function is

$A\:e^{2x}+B\:e^{-x}$

Hence the solution is

$y=A\:e^{2x}+B\:e^{-x}$