Question

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation. \frac{dy}{dx}=4e^{-3x}

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asked 2021-03-21
Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)

Expert Answers (1)

2021-03-23
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,
\(\displaystyle\int{\left.{d}{y}\right.}=\int{4}{e}^{{-{3}}}.{\left.{d}{x}\right.}\)
\(\displaystyle{y}={4}{\left[{\frac{{{e}^{{-{3}{x}}}}}{{-{3}}}}\right]}+{C}\)
\(\displaystyle{y}=-{\frac{{{4}}}{{{3}}}}{e}^{{-{3}{x}}}+{C}\)
\(\displaystyle{3}{y}=-{4}{e}^{{-{3}{x}}}+{3}{C}\)
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