Solved example of first order 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
Integrate both sides, the left side with respect to $y$, and the right side with respect to $x$
The integral of a constant by a function is equal to the constant multiplied by the integral of the function
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function
Solve the integral $\int4ydy$ and replace the result in the differential equation
The integral of a constant by a function is equal to the constant multiplied by the integral of the function
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function
Take $\frac{5}{3}$ out of the fraction
Solve the integral $\int5x^2dx$ 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 $C$
Divide both sides of the equation by $2$
Removing the variable's exponent
As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\frac{\sqrt{\frac{5}{3}x^{3}+C_0}}{\sqrt{2}}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
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