- A-Level Further Maths

# 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 isMove 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 aboveMultiply 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 sideRearrange to make y the subject - this is the

**general solution**Find the

**particular solution**by substituting in boundary conditions (if there are any)