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A first order differential equation y = f(x, y) is a linear equation if the function f is a “linear” expression in y. That is, the equation is linear if the 1.2 First Order Ordinary Differential Equations (ODE). Types of first order ODE. - Separable equation. - Homogenous equation. - Exact equation. Separation of variables is a technique commonly used to solve first order ordinary differential equations. It is so-called because we rearrange the equation Direct Method of solving linear first-order ODE's. • Examples. Page 3. Definition. 1. Separable equations. 3. Homogeneous equations. 4. Modeling with first order differential equations. 5. Differences between linear and nonlinear equations. A first-order linear differential equation is one that can be put into the form Notice that this differential equation is not separable because it's Solution Equation (5) is a first-order linear differential equation for i as a function of t. Its standard form is. (6) di dt. +. R.