Question

Solve the Initial Value problem
y"− 4y′ + 3y = 0,y( 0 )= 7,y′ (0) = 11

Answer 1

$$y''-4y'+3y=0$$

The characteristic equation is

$$r^2-4r+3=0$$

$$(r-1)(r-3)=0$$

$$r_1=1, r_2=3$$

The general solution of the homogeneous differential equation is

$$y(t)=c_1e^t+c_2e^{3t}$$

$$y(0)=7: 7=c_1e^0+c_2e^{3(0)}$$

$$c_1+c_2=7$$

$$y'(t)=c_1e^t+3c_2e^{3t}$$

$$y'(0)=11: 11=c_1e^0+3c_2e^{3(0)}$$

$$c_1+3c_2=11$$

Then

$$c_1=5, c_2=2$$

Solution of the Initial Value problem is

$$y(t)=5e^t+2e^{3t}$$