Homogeneous Linear Equations


Part e: Cauchy-Euler Equations


Each differential equation two fundamental solutions of the form tnt^n.

For every equation, we only consider values where t>0t > 0.


Your solution family 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.

t2y′′−3ty′+4y=0t^2y'' - 3ty' + 4y = 0

Initial Conditions to Check:

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