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.

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.