Editing Separation of variables (section)

=== Example ===
Consider the simple nonlinear second-order differential equation:<math display="block">y''=(y')^2.</math>This equation is an equation only of ''y<nowiki>''</nowiki>'' and ''y'<nowiki/>'', meaning it is reducible to the general form described above and is, therefore, separable. Since it is a second-order separable equation, collect all ''x'' variables on one side and all ''y'<nowiki/>'' variables on the other to get:<math display="block">\frac{d(y')}{(y')^2}=dx.</math>Now, integrate the right side with respect to ''x'' and the left with respect to ''y''':<math display="block">\int \frac{d(y')}{(y')^2}=\int dx.</math>This gives<math display="block">-\frac{1}{y'}=x+C_1,</math>which simplifies to:<math display="block">y'=-\frac{1}{x+C_1}~.</math>This is now a simple integral problem that gives the final answer:<math display="block">y=C_2-\ln|x+C_1|.</math>