Question

In Problems 1–22 solve the given differential equation by separation

of variables. $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{5}\mathit{x}$

Short answer

The solution of the given differential equation is $y=-\frac{1}{5}\cos 5x+C$.

Step-by-step solution

Separable Equation

A first-order differential equation of the form $\frac{dy}{dx}=g\left(x\right)h\left(y\right)$Step 2: Separate the variables and integrate is said to be separable or to have separable variables.

Separate the variables and integrate

The given equation is, $\frac{dy}{dx}=\sin 5x$

Separate the variables,$dy=\sin 5xdx$

Integrate both sides,

$$\begin{gathered} \int dy=\int \sin 5xdx \\ \int dy=\frac{1}{5}\int \sin 5x\left(5\right)dx \end{gathered}$$

Let,

$$\begin{gathered} u=\sin 5x \\ du=5dx \end{gathered}$$

Where, $\sin udu=-\cos u+C$, so

$y=\frac{1}{5}(-\cos 5x)+C$

Simplify,

$y=-\frac{1}{5}\cos 5x+C$

Hence, the solution of the given differential equation is $\mathit{y}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{5}}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{5}\mathit{x}\mathbf{+}\mathit{C}$.