Differential Equations - Fundamental Set of Solutions

Question

Find the fundamental set of solutions for the given differential equation L[y]=y′′−9y′+20y=0 and initial point t0=0 that also specifies y1(t0)=1, y′1(t0)=0, y2(t0)=0 and y′2(t0)=1.

Answer 1

y'' - 9y' + 20 = 0

 

The characteristic equation is

 

λ² - 9λ + 20 = 0

 

(λ - 4)(λ - 5) = 20

 

λ = 4 or 5

 

General solution:

 

y(t) = ae^4t + be^5t

 

I am not sure what you mean by y₁ and y₂.  Generally, the initial conditions are specified for the entire solution, as

 

y(0) = ?

y'(0) = ?

 

These 2 initial conditions give you 2 equations in 2 unknowns a and b, which you can use to solve for a and b once you have the general solution.