Separable Equations - Department of Mechanical Engineering可分离的方程-机械工程系.docVIP

III. Ordinary Differential Equations (ODEs) (A) First order equations A linear, first order ODE is always solvable by the method of integrating factor: . Multiplying by (the integrating factor) will turn the left hand side to a perfect differential. ( To solve . Multiply by exp (3x) ( or (C is a constant) , or . Solve the following ODEs (a) (b) (The y term is on the other side. Be careful.) (c) (Remember to divide by the coefficient of first) Several families of nonlinear first order ODEs can be solved exactly. The equation , where n is real, is called the Bernoulli’s equation. Use the substitution to solve the ODE. ( To solve With one eventually obtains . ( The equation is called the Riccati’s equation. If one particular solution Y(x) is known, the substitution will give and hence u can be solved. An elementary example is . Try and the complete solution is . (B) Separable equations When a first order equation is of the form solutions can be obtained by simple quadrature (or integration) or ( To solve or or (41) Use separation of variables to find the general solution. Then, obtain the particular solution satisfying the given initial condition. Sketch the graph of the solution, showing the key features, and label any key values. (a) . (y = log (x3+1) ) (b) (y = tan (x ( 0.62660) ) (c) () (C) Homogeneous first order equations (Homogeneous functions) A function is said to be homogeneous of degree k if for any (. For example, , is of degrees 2 but is of degree 0. (Homogeneous equation) The equation can be solved by the substitution if f is homogeneous. CAUTION: The word homogeneous is used in a different context later in the chapter. Be careful. (42) Use to find the general solution for each of the following equations. (a) (b) (D) Exact equations and integrating factors For equations of the form or the idea is to examine if that can be turned into an exact differential. If a