# Second-order first-degree autonomous differential equation

## Contents

## Definition

Following the convention for autonomous differential equations, we denote the dependent variable by and the independent variable by .

### Form of the differential equation

A (one-dimensional and degree one) **second-order autonomous differential equation** is a differential equation of the form:

### Solution method and formula

We set a variable Then, we can rewrite . In particular, . Plug this in:

Solve this to obtain the general solution for in terms of . Plug this expression in:

and solve this first-order differential equation. Note that if is not expressible as an explicit function of , but we instead have a relational solution , then solve the first-order differential equation:

## Particular cases

### Case where the function on the right depends only on and not on

Consider a situation of the form:

We do the same substitution and obtain:

This is now a separable differential equation relating and . Integrate and obtain:

We thus get:

In particular, if is an antiderivative for , then we get:

where is a parameter. Each choice of gives a different solution.

Plug this back in and get:

(The indicates that there are in fact two differential equations and we need to take the union of their solution sets).

This is a first-order autonomous differential equation, and in particular a separable differential equation. Rearrange and get:

An additional constant, , arises from this indefinite integration. The upshot is that the general solution relates to and has two parameters , as we might expect from the degree of the equation.

### Case where the function on the right is multiplicatively separable

Consider a situation of the form:

We do the same substitution and obtain:

This is a separable differential equation and we can rearrange it to obtain:

We now perform the integration both sides. Suppose and . We get:

If can locally be inverted, we can write as an explicit function of We now plug this into the original differential equation and get:

This is now a first-order differential equation. If can locally be inverted, we can write as an explicit function of (locally) and then solve the resultant separable differential equation. Otherwise, there may be some other method available.

## Example

### Separable example

Consider the following differential equation:

We describe how to solve this using the technique discussed here. First, set . We get:

Thus, we get:

The case solves to gives

The case:

solves to give:

Now, we plug back the original substitution to get:

We now make cases based on the sign of .

#### Case

In this case, the differential equation becomes:

This has a stationary solution (which, incidentally, is *already* included in the prior solution family of all constant solutions) and the general solution can be obtained as:

This gives:

This solves to give:

To verify that this is a solution, we compute, using the chain rule for differentiation and differentiation rule for power functions:

and:

We can now plug these into the original differential equation and verify.

#### Case

Let . The differential equation becomes:

There are no stationary solutions. The general solution is given by:

This solves to give:

Rearranging, we get:

Note that a priori, applying tan to both sides causes us to lose the information that is in the range of . However, this constraint is "artificial" anyway and can be got rid of by readjusting the value of . Note that arc tangent constraints are artificial, but arc sine constraints are genuine, because the sine function repeats itselfwithina period but the tangent function does not.

To verify that this is a solution, we can compute:

We can now plug in and verify the original differential equation.

#### Case

Let . The differential equation becomes:

There are stationary solutions corresponding to and . However, all stationary solutions are already included in another solution family, so we can ignore these. We continue solving:

We get:

This rearranges to gives:

Exponentiate both sides to get:

The solutions are respectively:

#### All cases

The following are the solution families:

- (this is a singular solution family)