Supplementary Material Linaer & Non-Linear DE Dr. lory liza d. bulay-og, PECE Associate Prof 1 Linearity + The important issue is how the unknown y appears in the equation. A linear equation involves the dependent variable (y) and its derivatives by themselves. There must be no "unusual" nonlinear functions of y or its derivatives. + A linear equation must have constant coefficients, or coefficients which depend on the independent variable (t). If y or its derivatives appear in the coefficient the equation is non-linear. + A linear differential equation can be recognized by its form. It is linear if the coefficients of y (the dependent variable) and all order derivatives of y, are functions of t, or constant terms, only are all linear. Linearity - Examples The dependent variable is y, the equation contains Only the derivative of y with respect to t and Constant y, thus, it’s linear dy y0 dt The dependent variable is x, the equation contains 𝑥 2 , Thus non-linear dx x2 0 dt is linear is non-linear Linearity - Examples The dependent variable is y, the equation contains Only the derivative of y with respect to t and Constant y, thus, it’s linear The dependent variable is y, the equation contains y multiplied by derivative of y with respect to t, Thus non-linear dy 2 t 0 dt y dy 2 t 0 dt is linear is non-linear More examples: + + + + 𝑑𝑦 𝑥 + 2𝑦 = 2𝑥 𝑑𝑥 𝑑𝑦 + 𝑦𝑐𝑜𝑡𝑥 = 2𝑥 2 𝑑𝑥 𝑑𝑦 + 𝑥2𝑦 = 𝑥 𝑑𝑥 1 𝑑2 𝑦 3 = 3𝑥 − 𝑦 𝑥 𝑑𝑥 2 𝑑𝑦 (both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear) 𝑑𝑦 (both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear) 𝑑𝑦 (both 𝑑𝑥 𝑎𝑛𝑑 𝑦 𝑎𝑟𝑒 linear, thus the equation is linear) (the term 𝑦 3 is non linear, thus the equation is not linear) + 𝑑3 𝑦 𝑑𝑥 3 𝑑2 𝑦 −2 2 𝑑𝑥 + 𝑑𝑦 𝑑𝑥 = 2𝑠𝑖𝑛𝑥 𝑑 3 𝑦 𝑑 2 𝑦 𝑑𝑦 (the term , & 𝑎𝑟𝑒 𝑎𝑙𝑙 linear, thus the equation is linear) 𝑑𝑥 3 𝑑𝑥 2 𝑑𝑥 + 𝑑𝑦 𝑑𝑥 − 𝑠𝑖𝑛𝑦 = −𝑥 ( the term sin y is non-linear, thus the equation is not linear) Linearity – Summary Linear 2y dy dt (2 3 sin t) y dy t dt Non-linear y2 or y sin( y ) dy dt (2 3 y 2 ) y dy dt 2 Thanks! Any questions? You can find me at: + lory.bulay-og@ustp.edu.ph 9