Solved example of differential equations
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 of the equality
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
The integral of a constant is equal to the constant times the integral's variable
Solve the integral $\int1dy$ and replace the result in the differential equation
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 section within the integral with a new variable (let's call it $u$), 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
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
Isolate $dx$ in the previous equation
Substituting $u$ and $dx$ in the integral and simplify
Take the constant $\frac{1}{5}$ out of the integral
Apply the integral of the sine function: $\int\sin(x)dx=-\cos(x)$
Replace $u$ with the value that we assigned to it in the beginning: $5x$
Solve the integral $\int\sin\left(5x\right)dx$ and replace the result in the differential equation
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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