ODE Theory: General Solution to y'' + 4y' + 4y = 0

[smerhej]

Homework Statement

We have y'' + 4y' + 4y = 0 ; find the general solution.

Homework Equations

Reduction of Order.

The Attempt at a Solution

So when determining the roots of the characteristic equation, -2 was a double root, and therefore we can't simply have c₁e^-2t + c₂e^-2t. So I thought I would use reduction of order to get a second equation. However in the solution, they just left it c₁e^-2t + c₂e^-2t and I'm wondering if what I was taught to do in the case of non distinct roots was wrong, or if the solution is wrong.

 

[SteamKing]

The solution appears to be wrong.

y = c1*exp(-2t) + c2 * t * exp(-2t)

 

[HallsofIvy]

Please show exactly what you did in your attempted reduction of order. When I try a solution of the form $y= u(t)e^{-2t}$, I get $u(t)= A+ Bt$ giving $y= Ae^{-2t}+ Bte^{-2t}$ as general solution.