Homogeneous Linear Equations


Part a: Distinct Real Roots


Each differential equation has an associated characteristic equation with distinct real 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 factorization ar2+br+c=a(r−d)(r−e)ar^2 + br + c = a(r-d)(r-e), where d≠ed \neq e.


You must meet all initial conditions for the answer 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′′−5y′+6y=0y'' -5y' + 6y = 0

Initial Conditions to Check:

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