Question

Use one of our substitution techniques to solve: y''-y=0 Show that the solution can be written as:

y=c₁ eˣ+c₂ e⁻ˣ

Answer 1

Using the characteristic equation method, the differential equation y'' - y = 0 can be solved, leading to the general solution y = c1 e^x + c2 e^(-x), where c1 and c2 are constants.

Step-by-step explanation:

The differential equation y'' - y = 0 is a second-order linear homogeneous differential equation. To solve this, one can use the characteristic equation method. The characteristic equation for this differential equation is r^2 - 1 = 0. Solving this, we get two roots, r = 1 and r = -1. Therefore, the general solution to the differential equation is a linear combination of the solutions corresponding to these roots. Hence, y = c_1 e^x + c_2 e^{-x}, where c_1 and c_2 are arbitrary constants that can be determined by initial conditions or boundary values.