Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 71', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 458 Ordinary Differential Equations . Bernoulli equation y x f x y g x ya. A Bernoulli equation has the form y x f x y g x ya a 0 1. For a 0 and a 1 it is a linear equation see Paragraph . The substitution z y1-a brings it to a linear equation z x 1 - a f x z 1 - a g x which is discussed in Paragraph . With this in view one can obtain the general integral y1-a Ce F 1 - a e F J eFg x dx where F 1 - a J f x dx. . Equation of the form xy x y f x g y x . The substitution u y x brings the equation to a separable equation x2u x f x g u see Paragraph . . Darboux equation. A Darboux equation can be represented as f y xlh y g x h x . Using the substitution y xz x and taking z to be the independent variable one obtains a Bernoulli equation which is considered in Paragraph g z -zf z x z xf z xa 1h z . Some other first-order equations integrable by quadrature are treated in Section . . Exact Differential Equations. Integrating Factor . Exact differential equations. An exact differential equation has the form f x y dx g x y dy 0 where f . dy dx The left-hand side of the equation is the total differential of a function of two variables U x y . The general integral U x y C where C is an arbitrary constant and the function U is determined from the system d dU dx f y g. Integrating the first equation yields U f x y dx Qfy while integrating the variable y is treated as a parameter . On substituting this expression into the second equation one identifies the function and hence U . As a result the general integral of an exact differential equation can be represented in the form f f y df y g xo n dy c 7x0 Jyo where x0 and y0 are any numbers. . First-Order Differential Equations 459 TABLE An integrating factor j j x y for some types of ordinary differential equations f dx gdy 0 where f f x y and g g x y . The subscripts x and y indicate the corresponding partial derivatives No. Conditions for f and g .
