tailieunhanh - Calculus and its applications: 5.7

"Calculus and its applications: " - Objective: solve differential equations, verify that a given function is a solution of a differential equation, solve differential equations using separation of variables, solve applications involving differential equations. | 2012 Pearson Education, Inc. All rights reserved Slide Differential Equations OBJECTIVE Solve differential equations. Verify that a given function is a solution of a differential equation. Solve differential equations using separation of variables. Solve applications involving differential equations. 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Solve Solving Certain Differential Equations We actually find solutions of certain differential equations when we find their antiderivatives or indefinite integrals. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 2: Solve , given that f (0) = 8. We first find the general solution. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 2 (concluded): Then we substitute (0, 8) for (x, y) to find C. Thus, the particular solution is Differential Equations p. 453, formula 12, there is no “+C” 2012 Pearson Education, Inc. All rights reserved Slide Example 3: Show that is a solution of We first find Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 3 (concluded): Then we substitute into the differential equation: Since a true equation, 0 = 0, results, we know that is a solution to the differential equation. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Differential Equations Quick Check 1 Show that is a solution of 2012 Pearson Education, Inc. All rights reserved Slide Example 4: Solve We first separate the variables. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 4 (continued): We then integrate both sides. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 4 (concluded): Finally, we substitute x = 0 and y = 5 to find C. Thus, the particular solution is Differential . | 2012 Pearson Education, Inc. All rights reserved Slide Differential Equations OBJECTIVE Solve differential equations. Verify that a given function is a solution of a differential equation. Solve differential equations using separation of variables. Solve applications involving differential equations. 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Solve Solving Certain Differential Equations We actually find solutions of certain differential equations when we find their antiderivatives or indefinite integrals. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 2: Solve , given that f (0) = 8. We first find the general solution. Differential Equations 2012 Pearson Education, Inc. All rights reserved Slide Example 2 (concluded): Then we substitute (0, 8) for (x, y) to find C. Thus, the particular solution is Differential Equations p. 453, formula 12, there is no “+C” 2012 Pearson .