Equations expressible as separable

Some differential equations can made separable by using a suitable substitution.

i) ODEs of the form \( y'(x)= f\Big(\frac{y(x)}{x}\Big). \)

Example 7.

Let us solve the differential equation \( y'= \frac{x+y}{x-y}. \) The equation is not separable in this form, but we can make if separable by substituting \( u = \frac{y}{x}, \) resulting to \( y' = u + xu'. \) We get

\( u + xu'= \displaystyle \frac{1+u}{1-u}. \)

Separating the variables and integrating both sides, we get

\( \displaystyle \int \frac{1-u}{1+u^2} \, \mathrm{d}u= \int \frac{1}{x} \, \mathrm{d}x \)

\( \arctan{u} - \displaystyle \frac{1}{2} \ln(u^2 +1)= \ln{x} + C. \)

Substituting \( u = \frac{y}{x} \) and simplifying yields

\( \displaystyle \arctan{\frac{y}{x}} = \ln{C\sqrt{x^2 + y^2}}. \)

Here, it is not possible to derive an expression for y so we have to make do with just the implicit solution. The solutions can be visualized graphically:

ii) ODEs of the form \(\displaystyle y' = f(ax+by+c) \)

Another type of differential equation that can be made separable are equations of the form

\(\displaystyle y' = f(ax+by+c).\)

To rewrite the equation as separable, we use the substitution \(\displaystyle u = ax+by+c. \)

Example 8.

Let us find the solution to the differential equation

\(\displaystyle y' =(x-y)^2 +1. \)

Here, a natural substitution is \(\displaystyle u = x-y \Leftrightarrow y = x-u \Rightarrow y' = 1-u'. \) Substitution yields

\( \displaystyle 1-u' =u^2 +1 \)

\( \displaystyle \int -\frac{1}{u^2} \, \mathrm{d}u = \int \, \mathrm{d}x \)

\( \displaystyle \frac{1}{u} = x +C \)

\( \displaystyle y = x - \frac{1}{x +C}. \)