Homogeneous Linear Equations


Part b: Complex Conjugate Roots


Each differential equation has an associated characteristic equation with complex conjugate roots.


For example, an eqaution like

ay′′+by′+cy=0ay'' + by' + cy = 0
will have characteristic eqaution
ar2+br+c=0ar^2 + br + c = 0
with b2−4ac<0b^2 - 4ac < 0.


Your solutions must meet all initial conditions to be complete.

Please enter any multiplication explicitly with * symbols so the parcer can understand your answer, like this:

2t3e5t2⟶2∗t3∗e(5∗t2).2t^3e^{5t^2} \longrightarrow 2*t^3*e^{(5*t^2)}.

Problem 1

Find a family of functions that solves the equation and satisfies the conditions below.

y′′+y=0y'' + y = 0

Initial Conditions to Check:

  • ⬜️y(0) = 1 and y'(0) = 0
  • ⬜️y(0) = 0 and y'(0) = 1