Solving First Order Ordinary Differential Equations

Separable equation

Separable if and functions can be separated into separate one-variable functions (as shown below).

Homogenous equation

A function is homogenous when .

To solve:

• Use substitution, where is a function of and
• Differentiate both sides:
• Apply the substitution to make it separable

Reduction to homogenous type

This type of equation can be reduced to homogenous form.

If , use the substitution: .

In other cases:

• Find and such that and
• Use substitutions:

The reduced equation would be:

Linear equation

The above form is called the standard form.

The equation would be separable if .

Otherwise:

• Identify from the standard form
• Calculate integrating factor: . Integrate . Put it as the power of
• Multiply both sides by
• becomes
• Integrate both sides to solve for

Bernoulli’s equation

When or , the equation would be linear.

Otherwise, it can be converted to linear using as substituion.

None of the above

The equation must be converted to one of the above by using a suitable substitution.