Final Answer
Step-by-step Solution
Problem to solve:
Take $\frac{2}{3}$ 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 of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
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 $2$
Solve the integral $\int y^2dy$ 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
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 $\int\frac{2}{3}xdx$ 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 $C$
Simplify the fraction
Eliminate the $\frac{1}{3}$ from the left, multiplying both sides of the equation by $$
Solve the product $3\left(\frac{1}{3}x^2+C_0\right)$
We can rename $3C_0$ as other constant
Removing the variable's exponent
Find the explicit solution to the differential equation
