Solved example of first order differential equations
Take $\frac{5}{4}$ out of the fraction
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 $x$
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Solve the integral $\int ydy$ 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 number or constant function, such as $2$
Simplify the fraction $\frac{5}{4}\left(\frac{x^{3}}{3}\right)$
Solve the integral $\int\frac{5}{4}x^2dx$ 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 $C$
Eliminate the $\frac{1}{2}$ from the left, multiplying both sides of the equation by $$
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 $\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign
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