Question

Find all solution of the liner DE $\mathit{f}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\left(t\right)\mathbf{+}\mathbf{3}\mathit{f}\mathbf{\text{'}}\left(t\right)\mathbf{3}\mathit{f}\mathbf{\text{'}}\left(t\right)\mathbf{+}\mathit{f}\left(t\right)\mathbf{=}\mathbf{0}$.

Short answer

The solution is$f\left(t\right)=e^{-t}\left(c_1+c_{2}t+c_3{t}^2\right)$ .

Step-by-step solution

Determine the auxiliary equation of f'''(t)+3f'(t)3f'(t)+f(t)=0 .

Consider the linear differential equation$f\text{'}\text{'}\text{'}\left(\mathrm{t}\right)+3f\text{'}\left(\mathrm{t}\right)3f\text{'}\left(\mathrm{t}\right)+f\left(\mathrm{t}\right)=0$ .

Assume $D^n=f^n\left(t\right)$, substitute the value $D^3$for $f\text{'}\text{'}\left(t\right)$, $D^2$for$f\text{'}\text{'}\left(t\right)$ , D for$f\text{'}\left(t\right)$ and 1 for$f\left(t\right)$ in the equation$f\text{'}\text{'}\text{'}\left(t\right)+3f\text{'}\text{'}\left(t\right)+3f\text{'}\left(t\right)+f\left(t\right)=0$ as follows.

$$\begin{gathered} f\text{'}\text{'}\text{'}\left(t\right)+3f\text{'}\text{'}\left(t\right)+3f\text{'}\left(t\right)+f\left(t\right)=0 \\ {D}^3+3D^2+3D+1=0 \\ {\left(D+1\right)}^3=0 \end{gathered}$$

Determine the solution of (D+1)3=0 .

The solution of the differential equation${\left(D\pm a\right)}^n=0$ isrole="math" localid="1660805138475" $f\left(t\right)=e^{\sout{\mathbf{T}}\mathbf{at}}\left\{\sum _{i=0}^{n-1}{c}_{i+1}{t}^i\right\}$ .

Using the definition, the solution of the linear differential equation${\left(D+1\right)}^3=0$ is defined as follows.

$$\begin{gathered} f\left(t\right)=e^{-t}\left\{\sum _{i=0}^{n-1}{c}_{i+1}{t}^i\right\} \\ f\left(t\right)=e^{-t}\left\{c_1{t}^0+c_2{t}^1+c_3{t}^2\right\} \\ f\left(t\right)=e^{-t}\left\{c_1+c_2{t}^1+c_3{t}^2\right\} \end{gathered}$$

Hence, the solution of the linear differential equation$f\text{'}\text{'}\text{'}\left(t\right)+3f\text{'}\text{'}\left(t\right)+3f\text{'}\left(t\right)+f\left(t\right)=0$ is

$f\left(t\right)=e^{-t}\left\{c_1+c_2{t}^1+c_3{t}^2\right\}$