How do i solve differential equations?

To solve differential equations we use a method called separation of variables. This is where we take all the ‘y’ values to one side of the equation and all the ‘x’ values to the other side of the equation make sure the ‘y’ terms are on the same side as the ‘dy/dx’. We then integrate both sides of the equation with respect to the variable of that side. We then if possible rearrange to equation with respect to y.

 

For example:

Solve this differential equation.

 

dy/dx = (3x²)/(y+1)

 

Step 1: Rearrange the equation so that the ‘y’ terms are on one side of the                                    equation and the ‘x’ terms are on the other side of the equation.

 

Times both sides by: (y+1).

 

(y+1)dy/dx = 3x²

 

Step 2: Integrate both sides with respect to x.

 

∫ (y+1) dx dy/dx = ∫ 3x² dx

 

(The ‘dx’s cancel on the left side of the equation leaving ‘dy’, this means we integrate the left side with respect to ‘y’ now).

 

 

∫ (y+1) dy = ∫ 3x² dx

 

(y²/2) + y = x³  + c

 

(You only need one constant in the solution for differential equations.)

 

It is not possible to rearrange this equation with respect to y so we leave it as it is.