Find the general solution of the given higher-order differential equation. d3u dt3 + d2u dt2 − 2u = 0

According to the roots of the characteristic equation, the solution is given by:

In which A, B and C are constants related to the initial conditions.

The differential equation is:

The characteristic equation is:

It's roots are:

Thus, the solution is:

In which A, B and C are constants related to the initial conditions.

A similar problem is given at link

This ODE has characteristic equation

which has roots at . Then the characteristic solution to the ODE is

answer: sine

step-by-step explanation:

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1