Simply put, a differential equation is said to be **separable** if the variables can be separated. That is, a separable equation is one that can be written in the form

Once this is done, all that is needed to solve the equation is to integrate both sides. The method for solving separable equations can therefore be summarized as follows: *Separate the variables and integrate*.

**Example 1**: Solve the equation 2 *y dy* = ( *x* ^{2} + 1) *dx*.

Since this equation is already expressed in “separated” form, just integrate:

**Example 2**: Solve the equation

This equation is separable, since the variables can be separated:

The integral of the left‐hand side of this last equation is simply

The solution of the differential equation is therefore

**Example 3**: Solve the IVP

The equation can be rewritten as follows:

Integrating both sides yields

Since the initial condition states that *y* = 1 at *x* = 0, the parameter *c* can be evaluated:

The solution of the IVP is therefore

**Example 4**: Find all solutions of the differential equation ( *x* ^{2} – 1) *y* ^{3} *dx* + *x* ^{2} *dy* = 0.

Separating the variables and then integrating both sides gives

Although the problem seems finished, there is another solution of the given differential equation that is not described by the family ½ *y* ^{−2} = *x* ^{−1} + *x* + *c*. In the separation step marked (†), both sides were divided by *y* ^{3}. This operation prevented *y* = 0 from being derived as a solution (since division by zero is forbidden). It just so happens, however, that *y* = 0 *is* a solution of the given differential equation, as you can easily check (note: *y* = 0 ⇒ *dy* = 0).

Thus, the complete solution of this equation must include

The lesson is clear:

If both sides of a separable differential equation are divided by some function *f*( *y*) (that is, a function of the dependent variable) during the separation process, then a valid solution may be lost. As a final step, you must check whether the constant function *y* = *y* _{0} [where *f*( *y* _{0}) = 0] is indeed a solution of the given differential equation. If it is, and if the family of solutions found by integrating both sides of the separated equation does not include this constant function, then this additional solution must be separately stated to complete the problem.

**Example 5**: Solve the equation

Separating the variables gives

*y* ^{2} – 1. Thus, the constant functions *y* = 1 and *y* = ‐1 may be lost as possible solutions; this will have to be checked later.) Integrating both sides of the separated equation yields

Now, both constant functions *y* = 1 and *y* = –1 *are* solutions of the original differential equation (as you can check by simply noting that *y* = ±1 ⟹ *dy*/ *dx* = 0), and neither is described by the family above. Thus the complete set of solutions of the given differential equation includes

**Example 6**: Solve the differential equation *xydx* – ( *x* ^{2} + 1) *dy* = 0.

Separate the variables,

Note that in the separation step (†), both sides were divided by *y*; thus, the solution *y* = 0 may have been lost. Direct substitution of the constant function *y* = 0 into the original differential equation shows that it is indeed a solution. However, the family *y* ^{2} = *c*( *x* ^{2} + 1) already includes the function *y* = 0 (take *c* = 0), so it need not be separately mentioned.

**Example 7**: Find the curve *r* = *r*(θ) in polar coordinates that solves the IVP

The given equation is separable, since it can be expressed in the separated form

Now integrate both sides:

Since the solution curve is to pass through the point with polar coordinates ( *r*, θ) = (2, π),

The solution of the IVP is therefore

This is a circle of diameter 2, tangent to the *y* axis at the origin; see Figure *r* (which is the dependent variable here). However, even though *r* = 0 formally satisfies the differential equation, it clearly does not satisfy the initial condition *r*(π) = 2.

**Figure 1**