Question

Determine the values of r for which the differential equation y′′′+9y′′+20y′=0 has solutions of the form y=e^rt. Enter the values of r in increasing order. If there is no answer, enter DNE.

r = ___

r = ___

r = ___

Answer 1

Now the given initial value problem is   y’’’+9y’’+20y’ = 0 -----(1)

Now let us find the characteristic equation for homogeneous equation by assuming the solution y = e^rt which is not equal to Zero.

So y’ = r e^rt

      y’’ = r²e^rt     

and y’’’ = r³e^rt so substitute in (1)

r³e^rt + 9r²e^rt + 20e^rt = 0

e^rt (r³ + 9r² + 20r) = 0

e^rt is not equal to zero so r³ + 9r² + 20r = 0

• r (r² + 9r + 20) = 0
• r (r² + 4r + 5r + 20) = 0
• r (r (r + 4) + 5(r +4)) = 0
• r (r + 4) ( r + 5) = 0
• so r values will be r₁ = 0 ,r₂ = -4 and r₃ = -5
• so when writing in increasing order
• r₁ = -5
• r₂ = -4
• r₃ = 0