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1

Solved example of differential equations

$\frac{dy}{dx}=\sin\left(5x\right)$
2

Multiply both sides of the equation by $dx$

$dy=\sin\left(5x\right)\cdot dx$
3

Integrate both sides, the left side with respect to $y$, and the right side with respect to $x$

$\int1dy=\int\sin\left(5x\right)dx$

The integral of a constant is equal to the constant times the integral's variable

$y$
4

Solve the integral $\int1dy$ and replace the result in the differential equation

$y=\int\sin\left(5x\right)dx$

Solve the integral $\int\sin\left(5x\right)dx$ applying u-substitution. Let $u$ and $du$ be

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

Isolate $dx$ in the previous equation

$\frac{du}{5}=dx$

Substituting $u$ and $dx$ in the integral and simplify

$\int\frac{\sin\left(u\right)}{5}du$

Take the constant out of the integral

$\frac{1}{5}\int\sin\left(u\right)du$

Apply the integral of the sine function

$-\frac{1}{5}\cos\left(u\right)$

Substitute $u$ back for it's value, $5x$

$-\frac{1}{5}\cos\left(5x\right)$
5

Solve the integral $\int\sin\left(5x\right)dx$ and replace the result in the differential equation

$y=-\frac{1}{5}\cos\left(5x\right)$
6

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$y=-\frac{1}{5}\cos\left(5x\right)+C_0$

Answer

$y=-\frac{1}{5}\cos\left(5x\right)+C_0$

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