Question

What is the solution of the associated homogeneous equation y'' + y = 0

Answer 1

Answer:
`y(t)=Acos(t)+Bsin(t)`

Explanation:

To find the solution of a given differential equation ay''+by'+cy=0, a≠0, you have to consider the quadratic polynomial ax²+bx+c=0, called the characteristic polynomial.

Using the quadratic formula, this polynomial will always have one or two roots, for example r and s. The general solution of the differential equation is:

`y(t)= Ae^(rt)+Be^(st)` , if the roots r and s are real numbers and r≠s.

`y(t)= A e^(rt)+B*t*e^(rt)` , if r=s is real.

`y(t)=Acos(\beta t)e^(\alpha t) +Bsin(\beta t)e^(\alpha t)` , if the roots r and s are complex numbers α+βi and α−βi

.

In this case, the characteristic polynomial is:

`x^(2) +1=0\\x^(2) =-1\\x1=i; x2=-i`

Since the roots are complex numbers, with α=0 and β=1, then the answer is:
`y(t)=Acos(t)+Bsin(t)`