Find the thirdorder differential equation having the functio

Find the third-order differential equation having the function as the general solution.

y = c₁ + c₂e^-2x + c₃e^7x

Solution

y = c₁ + c₂e^-2x + c₃e^7x

It can be re-written as

y = c₁ exp(0*x) + c₂e^-2x + c₃e^7x

We can see that the characteristic equation has 3 roots

r=0,r=-2 and r = 7

Hence the characteristic equation is

(r-0)*(r+2)*(r-7) = 0

r*(r^2-5r-14) = 0

r^3-5r^2-14r=0

Converting this to differential equation form we get

y\'\'\'-5y\'\'-14y\'=0