# determine two linearly independent solutions to the given differential equation

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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sovienesY
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.