x = c1 cos(t) + c2 sin(t) is a two-parameter family of solutions of the second-order DE x'' + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. x(0) = −1, x'(0) = 2

Step-by-step explanation:

The problem is very simple, since they give us the solution from the start. However I will show you how they came to that solution:

A differential equation of the form:

Will have a characteristic equation of the form:

Where solutions are the roots from which the general solution can be found.

For real roots the solution is given by:

For real repeated roots the solution is given by:

For complex roots the solution is given by:

Where:

Let's find the solution for using the previous information:

The characteristic equation is:

So, the roots are given by:

Therefore, the solution is:

As you can see, is the same solution provided by the problem.

Moving on, let's find the derivative of in order to find the constants and :

Evaluating the initial conditions:

And

Now we have found the value of the constants, the solution of the second-order IVP is:

correct answer is c. just did it

step-by-step explanation:

kayla would of sent 8

step-by-step explanation: