`dy/dx = 6x^2` Use integration to find a general solution to the differential equation

Expert Answers

An illustration of the letter 'A' in a speech bubbles

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...

See
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Get 48 Hours Free Access

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`

 

Approved by eNotes Editorial Team