monad Change the differential equation x 4x 12x 0 into a m

Change the differential equation x\" + 4x\'- 12x = 0 into a matrix form: X rightarrow = AX rightarrow where X rightarrow = (x_1 x_2), Find the general solution. Find the solution if x(0) = 2 and x\'(0) = 7. Draw the phase portrait by hand. State what kind of equilibrium point the origin is.

Solution

Given differential equation is

X11+4x¹-12x=0

Corresponding auxiliary equation (a.e) is

M2+4m-12=0

M2+6m-2m-12=0

M(m+6)-2(m+6)

(m+6)(m-2)=0

m=-6 or m=2

General solution is

x= c1 e-6x+c2 e2x

If x(0)=2

2=c1+c2--------(1)

x1=-6c1e^-6x+2c₂e^2x

if x1(0)=7

7=-6c1+2c2-----------(2)

By solving eq 1 and 2

C1=-3/8

C2= 19/8

X=-3/8 e-6x+ 19/8 e2x