3. Integrable Combinations

There are many differential equations where we cannot separate the variables, like we saw in the previous section.

However, we can possibly solve the DE if we use one of the following expressions to get the differential equation in a form that we can solve:

(1) `d(xy) = x dy + y dx`

(2) `d(x^2+ y^2) = 2(x dx + y dy)`

(3) `d(y/x)=(x dy-y dx)/x^2`

(4) `d(x/y)=(y dx-x dy)/y^2`

These differential forms are called integrable combinations. If we can transform our differential equation into one side of one of the above forms, then we can use the other side to solve the DE.

Where do these integrable combinations come from?

One way to think of the first expression above is as follows. Say x and y are both functions of t. If we have the product xy, then the derivative with respect to t of that product would be given by:


(This is just the product rule that we learned ages ago.) Then we can multiply throughout by dt and we get the expression in (1) above, `d(xy) = x dy + y dx`.

We saw the second combination, `d(x^2+ y^2) = 2(x dx + y dy)`, back in implicit differentiation and related rates.

The third and fourth combinations are based on the Quotient Rule.

Example 1


(2y + x)dy + y dx = 0

Here is the solution graph of our answer for Example 1 (I've used K = 25) . It is a hyperbola, with asymptotes y = −x, and the x-axis:

solution of differential equation - implicit function

Solution using `K=25`.

Example 2


x dx = 9x2 dx y dy

Here is the graph of our answer for Example 2 (using K = 9). It is an implicit function:

solution of differential equation - implicit function

Solution using `K=2`.