Differential Equations
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.
Integrating Factor
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)

At further stage, we also look at second-order differential equations - these come in two types: homogenous and non-homogenous.
Second-Order Homogeneous Differential Equations
Second-order homogeneous differential equations can be solved when in the form:

Homogeneous means it equals zero
The solution in terms of y will have four constants that need to be found:
λ and μ, which are found by solving the auxiliary equation
A and B, which can only be found if you have boundary conditions
The auxiliary equation is am² + bm + c = 0, and its solutions are λ and μ
a, b, and c are the coefficients in the second-order homogenous differential equation above.
The format of the solution in terms of y depends on the auxiliary equation:

Second-Order Non-Homogeneous Differential Equations
Second-order homogeneous differential equations can be solved when in the form:

The left hand side of this equation is known as the corresponding homogeneous equation and its solution is called the complementary function.
The right hand side of this equation is a function of x and its solution is called the particular integral.
Non-homogeneous means the corresponding homogeneous equation equals a function of x
To solve a second-order non-homogeneous differential equation, first solve the corresponding homogeneous equation as normal (see above), and then find the particular integral:
Take the correct standard function
Find dy/dx and d²y/dx²
Substitute these into the initial equation
Solve to find the constants in the standard function
The form of the standard function depends on the form of f(x):

When the particular integral is already in the complementary function, you need to multiply the particular integral by x
For example, if f(x) is in the exponential form where k is the same as one of the roots in the auxiliary equation, multiply the particular integral by x.