[理学]线性代数III-chapter62.pptVIP

Which is the general solution of the given linear equation. Example 12 Solution It seems that the equation does not belong the types we have learned. But if we regard the independent variable as y, and rewrite the equation as follows: Which is the general solution of the given linear equation. 4. Bernoulli’s equations Definition 4 Notations Solution method of Bernoulli equation Substituting these into the above equation, we obtain which is a linear equation. Example 13 Solution Equations solvable by transformations of variables Example 14 Solution Substituting it into the equation, we obtain Separating the variables, we obtain Integrating both sides, we obtain we obtain the general solution of the original equation It is easy to see that this equation can be transformed into a separable equation by substitution u=ax+by+c. Example 15 Solution The original equation becomes Thus, the general solution of the original equation is Example 16 Solution Thus, the general solution of the original equation is Notations (3) We have seen that except for a few particular equations, solving equations by transformations of variables is not very easy because it is hard to see in general how to find a suitable transformation. It requires some skills gained only by experience. 5. Total differential equations Definition 5 Solution method Example 17 Solution Method 1 (by line integrals) Method 2 ( indefinite integral method) Method 3 ( combining terms into total differentials) Example 18 Solution Example 19 Solution * Prof Liubiyu New Words First-Order Differential Equation 一阶微分方程 Equations with Variable Separable 可分离变量方程 Homogeneous Equations 齐次方程 Linear Equations of first order 一阶线性方程 Bernoulli Equations Bernoulli方程 Exact Equations (total differential equations) 全微分方程 The Method of Variation of Constants 常数变易法 Contents §6.2 First order differential equations §6.3 Differential equations of higher order solvable by reduced order m