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
asked 2021-09-17
Solve each of the differential equations in Table 1. Include the characterestic polynomial and its roots with your answer.
\(\displaystyle{y}{''}-{3}{y}'+{2}{y}={0}\)

Expert Answers (1)

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, \(\displaystyle{y}={c}_{{1}}{e}^{{x}}+{c}_{{2}}{e}^{{{2}{x}}}\)
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