Show that y c1e2x c2e3x where c1 c2 are arbitrary constant

Show that y = c_1e^2x + c_2e^3x, where c_1, c_2 are arbitrary constants, is the general solution of y\" - 5y\' + 6y = 0. (DO NOT SOLVE anything. Show and solve demand different actions as discussed earlier]

Solution

Given that

y\'\' - 5y\' + 6y = 0

d²y/dx² - 5 dy/dx + 6y = 0

The auxialary equation is ,

r² - 5r + 6 = 0

r² - 2r - 3r + 6 = 0

r(r - 2) - 3(r - 2) = 0

(r - 2)(r - 3) = 0

r - 2 = 0 , r - 3 = 0

r = 2 , r = 3

Let r₁ = 2 , r₂ = 3

If the roots are real and distinct then the general solution is ,

y = c₁e^r₁x + c₂e^r₂x

y = c₁e^2x + c₂e^3x

Therefore ,

     y = c₁e^2x + c₂e^3x ,where c₁,c₂ are arbitary constants , is the general solution of   y\'\' - 5y\' + 6y = 0