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1

Solved example of differential equations

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

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side

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

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

We can solve the integral $\int\sin\left(5x\right)dx$ by applying integration by substitution method (also called U-Substitution). First, we must identify a part of the integral with a new variable, which when substituted makes the integral easier. We see that $5x$ it's a good candidate for substitution. Let's define a variable $u$ and assign it to the choosen part

$u=5x$

Now, in order to rewrite $dx$ in terms of $du$, we need to find the derivative of $u$. We need to calculate $du$, we can do that by deriving the equation above

$du=5dx$

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 with the value that we assigned to it: $5x$

$-\frac{1}{5}\cos\left(5x\right)$
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 integrating we must add the constant of integration

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

Final Answer

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

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