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C.2 Separation of Variables Use Separation of Variables to Solve Differential Equations

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C.2 Separation of Variables Use Separation of Variables to Solve Differential Equations APPENDIX C Differential Equations A27 C.2 Separation of Variables Use separation of variables to solve differential equations. • Use differential equations to model and solve real-life problems. Separation of Variables The simplest type of differential equation is one of the form yЈ ϭ f ͑x͒. You know that this type of equation can be solved by integration to obtain y ϭ ͵ f ͑x͒ dx. In this section, you will learn how to use integration to solve another important family of differential equations—those in which the variables can be separated. TECHNOLOGY This technique is called separation of variables. You can use a symbolic integration utility to solve a separable variables differential Separation of Variables equation. Use a symbolic integra- If f and g are continuous functions, then the differential equation tion utility to solve the differential equation dy ϭ f ͑x͒g͑y͒ x dx yЈ ϭ . y2 ϩ 1 has a general solution of 1 ͵ dy ϭ ͵ f ͑x͒ dx ϩ C. g͑y͒ Essentially, the technique of separation of variables is just what its name implies. For a differential equation involving x and y, you separate the x variables to one side and the y variables to the other. After separating variables, integrate each side to obtain the general solution. Here is an example. EXAMPLE 1 Solving a Differential Equation dy x Find the general solution of ϭ . dx y2 ϩ 1 Solution Begin by separating variables, then integrate each side. dy x ϭ Differential equation dx y2 ϩ 1 ( y2 ϩ 1͒ dy ϭ x dx Separate variables. ͵͑y2 ϩ 1͒ dy ϭ ͵x dx Integrate each side. y 3 x2 ϩ y ϭ ϩ C General solution 3 2 A28 APPENDIX C Differential Equations EXAMPLE 2 Solving a Differential Equation Find the general solution of dy x ϭ . dx y Solution Begin by separating variables, then integrate each side. dy x ϭ Differential equation dx y y dy ϭ x dx Separate variables. ͵y dy ϭ ͵x dx Integrate each side. y2 x2 ϭ ϩ C Find antiderivatives. 2 2 1 y2 ϭ x2 ϩ C Multiply each side by 2. 2 ϭ 2 ϩ So, the general solution is y x C. Note that C1 is used as a temporary constant of integration in anticipation of multiplying each side of the equation by 2 to produce the constant C. STUDY TIP After finding the general solution of a differential equation, you should use the techniques demonstrated in Section C.1 to check the solution. For instance, in Example 2 you can check the solution by differentiating the equation y 2 ϭ x 2 ϩ C to obtain 2yyЈ ϭ 2x or yЈ ϭ x͞y. EXAMPLE 3 Solving a Differential Equation Find the general solution of dy ey ϭ 2x. dx Use a graphing utility to graph several solutions. Solution Begin by separating variables, then integrate each side. dy ey ϭ 2x Differential equation dx y e dy ϭ 2x dx Separate variables. 5 C 15 ͵ y ϭ ͵ Integrate each side. e dy 2x dx y 2 C 5 C 10 e ϭ x ϩ C Find antiderivatives. 6 6 By taking the natural logarithm of each side, you can write the general solution as C 0 y ϭ ln͑x2 ϩ C͒. General solution 5 The graphs of the particular solutions given by C ϭ 0, C ϭ 5, C ϭ 10, and FIGURE A.10 C ϭ 15 are shown in Figure A.10. APPENDIX C Differential Equations A29 EXAMPLE 4 Finding a Particular Solution Solve the differential equation xex2 ϩ yyЈ ϭ 0 subject to the initial condition y ϭ 1 when x ϭ 0. Solution x2 xe ϩ yyЈ ϭ 0 Differential equation dy 2 y ϭϪxex Subtract xex2 from each side. dx ϭϪ x2 y dy xe dx Separate variables. 2 ͵ ϭ ͵Ϫ x Integrate each side. y dy xe dx 2 y 1 2 ϭϪ ex ϩ C Find antiderivatives. 2 2 1 2 y2 ϭϪex ϩ C Multiply each side by 2. To find the particular solution, substitute the initial condition values to obtain ͑1͒2 ϭϪe͑0͒2 ϩ C. This implies that 1 ϭϪ1 ϩ C, or C ϭ 2. So, the particular solution that satisfies the initial condition is 2 y2 ϭϪex ϩ 2. Particular solution EXAMPLE 5 Solving a Differential Equation Example 3 in Section C.1 uses the differential equation dx ϭ k͑10 Ϫ x͒ dt to model the sales of a new product. Solve this differential equation. Solution dx STUDY TIP ϭ k͑10 Ϫ x͒ Differential equation dt In Example 5, the context of 1 the original model indicates that dx ϭ k dt Separate variables. 10 Ϫ x ͑10 Ϫ x͒ is positive. So, when ͑͞ Ϫ ͒ 1 you integrate 1 10 x , you ͵ dx ϭ ͵k dt Integrate each side. can write Ϫln͑10 Ϫ x͒, rather 10 Ϫ x than ϪlnԽ10 Ϫ xԽ. Ϫ ͑ Ϫ ͒ ϭ ϩ ln 10 x kt C1 Find antiderivatives. Also note in Example 5 ͑ Ϫ ͒ ϭϪ Ϫ Ϫ ln 10 x kt C1 Multiply each side by 1. that the solution agrees with ϪktϪC 10 Ϫ x ϭ e 1 Exponentiate each side. the one that was given in Ϫ Example 3 in Section C.1. x ϭ 10 Ϫ Ce kt Solve for x. A30 APPENDIX C Differential Equations Applications EXAMPLE 6 Modeling National Income Let y represent the national income, let a represent the income spent on necessi- ties, and let b represent the percent of the remaining income spent on luxuries. A commonly used economic model that relates these three quantities is dy ϭ k͑1 Ϫ b͒͑y Ϫ a͒ dt where t is the time in years. Assume that b is 75%, and solve the resulting differ- ential equation. Corporate profits in the United Solution Because b is 75%, it follows that ͑1 Ϫ b͒ is 0.25. So, you can solve the States are closely monitored by differential equation as follows. New York City’s Wall Street execu- dy ϭ k͑0.25͒͑y Ϫ a͒ Differential equation tives. Corporate profits, however, dt represent only about 11.9% of 1 the national income. In 1999, the dy ϭ 0.25k dt Separate variables. y Ϫ a national income was more than $7 trillion. Of this, about 71% 1 ϭ ͵ Ϫ dy ͵0.25k dt Integrate each side. was employee compensation. y a ͑ Ϫ ͒ ϭ ϩ Ϫ ln y a 0.25kt C1 Find antiderivatives, given y a > 0. y Ϫ a ϭ Ce0.25kt Exponentiate each side. y ϭ a ϩ Ce0.25kt Add a to each side. The graph of this solution is shown in Figure A.11. In the figure, note that the national income is spent in three ways. ͑National income͒ ϭ ͑necessities͒ ϩ ͑luxuries͒ ϩ ͑capital investment͒ Modeling National Income y Capital investment Income consumed on Consumed on luxuries necessities and luxuries 0.25kt Consumed on necessities ya= + 0.75 Ce National income yaCe= + 0.25kt C a t 1234567891011121314 Time (in years) FIGURE A.11 APPENDIX C Differential Equations A31 EXAMPLE 7 Using Graphical Information Find the equation of the graph that has the characteristics listed below. 1. At each point ͑x, y͒ on the graph, the slope is Ϫx͞2y. 2. The graph passes through the point ͑2, 1͒. Solution Using the information about the slope of the graph, you can write the differential equation dy x ϭϪ . dx 2y Using the point on the graph, you can determine the initial condition y ϭ 1 when x ϭ 2. dy x ϭϪ Differential equation dx 2y 2 y dy ϭϪx dx Separate variables. ͵2y dy ϭ ͵Ϫx dx Integrate each side. x2 y2 ϭϪ ϩ C Find antiderivatives. 2 1 2 y2 ϭϪx2 ϩ C Multiply each side by 2. 2 ϩ 2 ϭ x 2y C Simplify. 2 Applying the initial condition yields ͑2͒2 ϩ 2͑1͒2 ϭ C 3 3 which implies that C ϭ 6. So, the equation that satisfies the two given conditions is x2 ϩ 2y2 ϭ 6. Particular solution 2 As shown in Figure A.12, the graph of this equation is an ellipse. FIGURE A.12 TAKE ANOTHER LOOK Classifying Differential Equations In which of the differential equations can the variables be separated? dy 3x a. ϭ dx y dy 3x b. ϭ ϩ 1 dx y dy 3x c. x2 ϭ dx y dy 3x ϩ y d. ϭ dx y A32 APPENDIX C Differential Equations The following warm-up exercises involve skills that were covered in earlier sections. WARM-UP C.2 You will use these skills in the exercise set for this section. In Exercises 1–6, find the indefinite integral and check your result by differentiating. 1. ͵ x3͞2 dx 2. ͵͑t3 Ϫ t1͞3͒ dt 2 3. ͵ dx x Ϫ 5 y 4. ͵ dy 2y2 ϩ 1 5. ͵e2y dy 6. ͵xe1Ϫx2 dx In Exercises 7–10, solve the equation for C or k. 7.͑3͒2 Ϫ 6͑3͒ ϭ 1 ϩ C 8. ͑Ϫ1͒2 ϩ ͑Ϫ2͒2 ϭ C 9.10 ϭ 2e2k 10. ͑6͒2 Ϫ 3͑6͒ ϭ eϪk EXERCISES C.2 In Exercises 1–6, decide whether the variables in the differential 19.͑2 ϩ x͒yЈ ϭ 2y 20. yЈ ϭ ͑2x Ϫ 1͒͑y ϩ 3͒ equation can be separated. 21.xyЈ ϭ y 22. yЈ Ϫ y͑x ϩ 1͒ ϭ 0 dy x dy x ϩ 1 1.ϭ 2. ϭ x x dy x2 ϩ 2 dx y ϩ 3 dx x 23.yЈ ϭ Ϫ 24. ϭ y 1 ϩ y dx 3y2 dy 1 dy x 3.ϭ ϩ 1 4.
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