# 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)

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 isam² +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.