Separable Differential Equations

Separable differential equations are a fundamental class of first-order ordinary differential equations that can be expressed as the product of a function of the independent variable and a function of the dependent variable: $\displaystyle \frac{dy}{dx} = g(x)h(y)$. This form allows the variables to be separated and integrated independently by rewriting the equation as $\displaystyle \frac{1}{h(y)} dy = g(x) dx$. Integrating both sides yields , where is the integration constant determined by initial conditions. These equations commonly model engineering phenomena such as growth, decay, and mixing processes. Mastery of separability enables quick identification and solution of these ODEs and lays the foundation for more advanced methods like integrating factors and exact equations. Separable equations play a crucial role in predicting behaviors in mechanical, electrical, and chemical systems.