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:
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)`
Combining the results, we get the general solution for differential equation: