Question

show that y=c1e^x+c2e^2x is the general solution of y''-3y'+2y=0 on any interval, and find the particular solution for which y(0)=-1 and y'(0)=1

Answer 1

$$y''-3y'+2y=0$$

Characteristic (auxiliary) equation

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

$$r_1=1, r_2=2$$

The general solution of $y''-3y'+2y=0$ on any interval is

$$y=c_1e^x+c_2e^{2x}$$

Given $y(0)=-1, y'(0)=1$

$$y'=c_1e^x+2c_2e^{2x}$$

Then

$$-1=c_1+c_2$$

$$1=c_1+2c_2$$

$$c_1=-3, c_2=2$$

The particular solution is

$$y=-3e^x+2e^{2x}$$