determine two linearly independent solutions to the given differential equation

emancipezN 2021-09-29 Answered
determine two linearly independent solutions to the given differential equation of the form \(\displaystyle{y}{\left({x}\right)}={e}^{{{r}{x}}}\), and thereby determine the general solution to the differential equation.
y''+7y'+10y=0

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Expert Answer

sovienesY
Answered 2021-09-30 Author has 10042 answers
Step 1
Consider the provided differential equation,
y''+7y'+10y=0
Find the general solution of the differential equation.
Step 2
First find the roots of the differential equation,
\(\displaystyle{r}^{{{2}}}+{7}{r}+{10}={0}\)
\(\displaystyle{r}^{{{2}}}+{5}{r}+{2}{r}+{10}={10}\)
r(r+5)+2(r+5)=0
r=-2,-5
Therefore, the general solution is \(\displaystyle{y}={c}_{{{1}}}{e}^{{{r}_{{{1}}}{t}}}+{c}_{{{2}}}{e}^{{{r}_{{{2}}}{t}}}\).
So,
\(\displaystyle{y}={c}_{{{1}}}{e}^{{-{2}{t}}}+{c}_{{{2}}}{e}^{{-{5}{t}}}\)
Hence.
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