Question

Find a general solution to the given differential equation. y"-y'-2y=0

Answer 1

To find the general solution to the given differential equation y'' - y' - 2y = 0, follow these steps: Assume a solution of the form y = e^(rt), simplify the equation, set the characteristic equation equal to zero and solve for r, and write down the general solution for the different cases.

Step-by-step explanation:

To find a general solution to the given differential equation:

y'' - y' - 2y = 0

1. Assume a solution of the form y = e^(rt) and substitute it into the equation.
2. Simplify the equation and factor out the exponential term.
3. Set the characteristic equation equal to zero and solve for r.
4. Depending on the roots of the characteristic equation, there are three possible cases: real and distinct roots, complex roots, and repeated roots.
5. For each case, write down the general solution using the appropriate exponential functions.

For example, if the characteristic equation has real and distinct roots r1 and r2, the general solution is y = c1e^(r1t) + c2e^(r2t). If the characteristic equation has complex roots a + bi and a - bi, the general solution is y = e^(at) (c1cos(bt) + c2sin(bt)). If the characteristic equation has a repeated root r, the general solution is y = (c1 + c2t)e^(rt).