Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 41

An ordinary differential equation (ODE) is differential equation for the derivative of a function of one variable. When an ODE is in a form of y'=f(x,y) , this is just a first order ordinary differential equation.
The y ' is the same as (dy)/(dx) therefor first order ODE can written in a form of (dy)/(dx) = f(x,y)
That is form of the given problem: (dy)/(dx) = 6x^2.
We may apply integration after we rearrange it in a form of variable separable differential equation: N(y) dy = M(x) dx .
By cross-multiplication, we can be rearrange the problem into: (dy) = 6x^2dx .
Apply direct integration on both sides:
int (dy) =int 6x^2dx .
For the left side, we may apply basic integration property:
int (dy)=y
For the right side, we may apply the basic integration property: int c*f(x)dx = c int f(x) dx .
int 6x^2dx =6int x^2dx
Then apply Power Rule for integration: int u^n du= u^(n+1)/(n+1)+C
6 int x^2dx = 6*x^(2+1)/(2+1)
= 6*x^3/3+C
= 2x^3+C

Combining the results, we get the general solution for differential equation:
y=2x^3+C