Question # Solve each of the differential equations in Table 1. Include the characterestic polynomial and its roots with your answer.y''-3y'+2y=0

Second order linear equations
ANSWERED Solve each of the differential equations in Table 1. Include the characterestic polynomial and its roots with your answer.
$$\displaystyle{y}{''}-{3}{y}'+{2}{y}={0}$$ 2021-09-18
We find the solution of the given homogeneous differential equation after solving characteristic polynomial.
Given homogeneous differential equation
$$\displaystyle{y}{''}-{3}{y}'+{2}{y}={0}$$
Its characteristic polynomial is given by $$\displaystyle{m}^{{2}}-{3}{m}+{2}={0}$$
$$\displaystyle\Rightarrow{m}^{{2}}-{2}{m}-{m}+{2}={0}$$
$$\displaystyle\Rightarrow{m}{\left({m}-{2}\right)}-{1}{\left({m}-{2}\right)}={0}$$
$$\displaystyle\Rightarrow{\left({m}-{2}\right)}{\left({m}-{1}\right)}={0}$$
This gives m=1,2
These are th roots of the characteristic polynomial
Finally, the polution of given differential equation general
is $$\displaystyle{y}={{c}_{{e}}^{{{1}\cdot{x}}}}+{c}_{{2}}{e}^{{{2}\cdot{x}}}$$, where x is on indepenelent variable
Hence, $$\displaystyle{y}={c}_{{1}}{e}^{{x}}+{c}_{{2}}{e}^{{{2}{x}}}$$