In single maths, first-order differential equations are the only ones looked at, and are solved by separating the variables:
When dy/dx = f(x) g(y), you can say ∫ 1/g(y) dy = ∫ f(x) dx
Move all the y terms to the left where the dy is
Move all the x terms to the right, including the dx
This allows you to integrate each side with respect to the variable on that side to solve the equation.
You only need to add the ' +c ' to one side.
Just like when integrating an indefinite function, the initial result is a general solution and could be anywhere along the y-axis (see section on indefinite integral functions above). To fond the particular solution, you need to know a coordinate point on the curve - sometimes this is called a boundary condition.
This is not covered in single maths, but is an important method of solving first-order differential equations where x and terms are multiplied by one another in one of the terms:
Rearrange to be in the form dy/dx + P(x)y = Q(x)
Find the integrating factor using the formula above
Multiply the dy/dx by the integrating factor and by y
Multiply the right hand side by the integrating factor & simplify (if you can)
The middle term, P(x)y disappears
Move the dx to the right hand side and integrate this side
Rearrange to make y the subject - this is the general solution
Find the particular solution by substituting in boundary conditions (if there are any)