Solve this second order differential equation:?

Question

d^2y/dx^2 + 4dy/dx + 3y = 0

Answer 1

This is a homogenious equation(meaning the right side is 0)

and coefficents are all constants..

Standard method of solving homogenious constant-coefficient

linear ODE is to set up "Characteristic Equation".

Characteristic equation in this case is :

m^2 + 4m + 3 = 0

(m + 3)(m + 1) = 0

This quadratic equation has solutions of m = -3 or -1

With this solution,  set up the Complimentary

solution:

y = c1*e^(-3x) + c2*e^(-x)

Answer 2

Assuming y = e^mx,

dy/dx = my

d^2y/dx^2 = m^2 y

So

m^2y + 4my + 3y = 0

(m^2 + 4m + 3) y = 0

m^2 + 4m + 3 = 0, y is not zero

(m+3)(m+1) = 0

m = -3 and m = -1

y = k1 e^-3 + k2 e^-1