Step 1: Given that to

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)

Step 2: Solve

The given differential equation is a type of Variable Separable differential equation.

So, to solve this type of differential equation we have,

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)

\(\displaystyle{\left.{d}{y}\right.}={4}{e}^{{-{3}{x}}}.{\left.{d}{x}\right.}\)

Integrating both sides we obtain,

\(\displaystyle\int{\left.{d}{y}\right.}=\int{4}{e}^{{-{3}}}.{\left.{d}{x}\right.}\)

\(\displaystyle{y}={4}{\left[{\frac{{{e}^{{-{3}{x}}}}}{{-{3}}}}\right]}+{C}\)

\(\displaystyle{y}=-{\frac{{{4}}}{{{3}}}}{e}^{{-{3}{x}}}+{C}\)

\(\displaystyle{3}{y}=-{4}{e}^{{-{3}{x}}}+{3}{C}\)

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)

Step 2: Solve

The given differential equation is a type of Variable Separable differential equation.

So, to solve this type of differential equation we have,

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)

\(\displaystyle{\left.{d}{y}\right.}={4}{e}^{{-{3}{x}}}.{\left.{d}{x}\right.}\)

Integrating both sides we obtain,

\(\displaystyle\int{\left.{d}{y}\right.}=\int{4}{e}^{{-{3}}}.{\left.{d}{x}\right.}\)

\(\displaystyle{y}={4}{\left[{\frac{{{e}^{{-{3}{x}}}}}{{-{3}}}}\right]}+{C}\)

\(\displaystyle{y}=-{\frac{{{4}}}{{{3}}}}{e}^{{-{3}{x}}}+{C}\)

\(\displaystyle{3}{y}=-{4}{e}^{{-{3}{x}}}+{3}{C}\)