Second-Order Linear Equations
For constant-coefficient equations, form the characteristic equation and solve for its roots.
$$ay''+by'+cy=0 \quad \Rightarrow \quad ar^2+br+c=0$$
For constant-coefficient equations, form the characteristic equation and solve for its roots.
Solve $y''-5y'+6y=0$ if $y(0)=2$ and $y'(0)=5$.
Characteristic equation:
Apply initial conditions: $C_1+C_2=2$ and $2C_1+3C_2=5$.
Thus $C_1=1$ and $C_2=1$.
Final answer: $y=e^{2x}+e^{3x}$.