Find the general solution of y'' - 3y' + 2y = 0

Find the general solution of $y'' - 3y' + 2y = 0$.

Step 1: Characteristic equation

Substituting $y=e^{rx}$: $r^2-3r+2=0.$

Factoring, $(r-1)(r-2)=r^2-3r+2$, confirmed by expansion, so: \[(r-1)(r-2)=0 \implies r_1=1,\quad r_2=2.\]

Step 2: Fundamental system

Real, distinct roots give

\[ y_1=e^x, \qquad y_2=e^{2x}. \]

Step 3: Wronskian

\[ W(y_1,y_2)=\begin{vmatrix}e^x & e^{2x}\\e^x & 2e^{2x}\end{vmatrix} = 2e^{3x}-e^{3x} = e^{3x} \neq 0, \]

confirming $y_1,y_2$ are linearly independent for all $x$.

Step 4: General solution

The solution is recorded in the result below.