Question

Solve the differential equation $\frac{\mathbf{d}\mathbf{x}}{\mathbf{d}\mathbf{t}}\mathbf{+}\mathbf{3}\mathit{x}\mathbf{=}\mathbf{7}$and find the solution of the differential equation.

Short answer

The solution is$f\left(t\right)=\frac{7}{3}+\frac{7}{3}{e}^{-3t}C$ .

Step-by-step solution

Definition of first order linear differential equation

Consider the differential equation$f\text{'}\left(t\right)-af\left(t\right)=g\left(t\right)$ where$g\left(t\right)$ is a smooth function and$\text{'}a\text{'}$ is a constant. Then the general solution will be $f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt$.

Determination of the solution

Consider the differential equation as follows.

$x\text{'}\left(t\right)+3x\left(t\right)=7$

Now, the differential equation is in the form as follows.

$f\text{'}\left(t\right)-af\left(t\right)=g\left(t\right)$, where$g\left(t\right)$ is a smooth function, then the general solution will be as follows.

$f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt$

Compute the calculation of the solution

Substitute the value$7$for$g\left(t\right)$and$-3$for in$f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dt$as follows.

$\begin{array}{l}f\left(t\right)=e^{at}\int e^{-at}g\left(t\right)dtA\\ f\left(t\right)=e^{-3t}\int e^{3t}\times 7\times dt\\ f\left(t\right)=7e^{-3t}\int e^{3t}dt\end{array}$

Using substitution method in the integral as follows.

$\begin{array}{c}y=3t\\ dy=3dt\\ \frac{dy}{3}=dt\end{array}$

Substitute the value$3t$for$y$and $\frac{dy}{3}$for$dt$in$f\left(t\right)=7e^{3t}\int e^{-3t}dt$as follows.

$\begin{array}{l}f\left(t\right)=7e^{-3t}\int e^{3t}dt\\ f\left(t\right)=7e^{-3t}\int e^y\frac{dy}{3}\\ f\left(t\right)=\frac{7}{3}{e}^{-3t}(e^y+C)\end{array}$

Now, undo the substitution as follows.

$\begin{array}{c}f\left(t\right)=\frac{7}{3}{e}^{-3t}(e^y+C)\\ f\left(t\right)=\frac{7}{3}{e}^{-3t}(e^{3t}+C)\\ =\frac{7}{3}{e}^{-3t}{e}^{3t}+\frac{7}{3}{e}^{-3t}C\\ f\left(t\right)=\frac{7}{3}+\frac{7}{3}{e}^{-3t}C\end{array}$

Hence, the solution for the linear differential equation$x\text{'}\left(t\right)+3x\left(t\right)=7$ is$f\left(t\right)=\frac{7}{3}+\frac{7}{3}{e}^{-3t}C$