Solve by separation of variables dy/dx= sin5x.

Here we Solve by separation of variables dy/dx= sin5x.

Given differential equation is:

(1) $\begin{equation*} \frac{dy}{dx} = \sin(5x) \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} dy = \sin(5x)dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int dy = \int \sin(5x)dx \end{equation*}$

Integrating the left-hand side, we get:

(4) $\begin{equation*} y = \int dy = y + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

For the right-hand side, we can use the substitution $u = 5x$ and $du/dx = 5$ to obtain:

(5) $\begin{equation*} \int \sin(5x)dx = \frac{1}{5}\int \sin(u)du = -\frac{1}{5}\cos(u) + C_2 \end{equation*}$

where $C_2$ is another constant of integration. Substituting back $u=5x$ , we have:

(6) $\begin{equation*} \int \sin(5x)dx = -\frac{1}{5}\cos(5x) + C_2 \end{equation*}$

Substituting this back into our original equation, we have:

(7) $\begin{equation*} y = -\frac{1}{5}\cos(5x) + C_2 + C_1 \end{equation*}$

where $C_1$ and $C_2$ are constants of integration.

Therefore, the solution to the differential equation $\frac{dy}{dx} = \sin(5x)$ is:

(8) $\begin{equation*} y = -\frac{1}{5}\cos(5x) + C \end{equation*}$

where $C = C_1 + C_2$ is the constant of integration.