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• 1. Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefﬁcients, that is, equations of the form 1 ayЉ ϩ byЈ ϩ cy ෇ G͑x͒ where a, b, and c are constants and G is a continuous function. The related homogeneous equation 2 ayЉ ϩ byЈ ϩ cy ෇ 0 is called the complementary equation and plays an important role in the solution of the original nonhomogeneous equation (1). 3 Theorem The general solution of the nonhomogeneous differential equation (1) can be written as y͑x͒ ෇ yp͑x͒ ϩ yc͑x͒ where yp is a particular solution of Equation 1 and yc is the general solution of the complementary Equation 2. Proof All we have to do is verify that if y is any solution of Equation 1, then y Ϫ yp is a solution of the complementary Equation 2. Indeed a͑y Ϫ yp ͒Љ ϩ b͑y Ϫ yp ͒Ј ϩ c͑y Ϫ yp ͒ ෇ ayЉ Ϫ aypЉ ϩ byЈ Ϫ byp ϩ cy Ϫ cyp Ј ෇ ͑ayЉ ϩ byЈ ϩ cy͒ Ϫ ͑aypЉ ϩ byЈ ϩ cyp ͒ p ෇ t͑x͒ Ϫ t͑x͒ ෇ 0 We know from Additional Topics: Second-Order Linear Differential Equations how to solve the complementary equation. (Recall that the solution is yc ෇ c1 y1 ϩ c2 y2 , where y1 and y2 are linearly independent solutions of Equation 2.) Therefore, Theorem 3 says that we know the general solution of the nonhomogeneous equation as soon as we know a particular solution yp . There are two methods for ﬁnding a particular solution: The method of undetermined coefﬁcients is straightforward but works only for a restricted class of functions G. The method of variation of parameters works for every function G but i0s usually more difﬁcult to apply in practice. The Method of Undetermined Coefficients We ﬁrst illustrate the method of undetermined coefﬁcients for the equation ayЉ ϩ byЈ ϩ cy ෇ G͑x͒ where G͑x) is a polynomial. It is reasonable to guess that there is a particular solution yp that is a polynomial of the same degree as G because if y is a polynomial, then ayЉ ϩ byЈ ϩ cy is also a polynomial. We therefore substitute yp͑x͒ ෇ a polynomial (of the same degree as G ) into the differential equation and determine the coefﬁcients. EXAMPLE 1 Solve the equation yЉ ϩ yЈ Ϫ 2y ෇ x 2. SOLUTION The auxiliary equation of yЉ ϩ yЈ Ϫ 2y ෇ 0 is r 2 ϩ r Ϫ 2 ෇ ͑r Ϫ 1͒͑r ϩ 2͒ ෇ 0 1
• 2. 2 ■ NONHOMOGENEOUS LINEAR EQUATIONS with roots r ෇ 1, Ϫ2. So the solution of the complementary equation is yc ෇ c1 e x ϩ c2 eϪ2x Since G͑x͒ ෇ x 2 is a polynomial of degree 2, we seek a particular solution of the form yp͑x͒ ෇ Ax 2 ϩ Bx ϩ C Then yp ෇ 2Ax ϩ B and ypЉ ෇ 2A so, substituting into the given differential equation, we Ј have ͑2A͒ ϩ ͑2Ax ϩ B͒ Ϫ 2͑Ax 2 ϩ Bx ϩ C͒ ෇ x 2 or Ϫ2Ax 2 ϩ ͑2A Ϫ 2B͒x ϩ ͑2A ϩ B Ϫ 2C͒ ෇ x 2 Polynomials are equal when their coefﬁcients are equal. Thus Figure 1 shows four solutions of the differential equation in Example 1 in terms of the particular solution yp and the functions f ͑x͒ ෇ e x and t͑x͒ ෇ eϪ2 x. ■ ■ 8 2A Ϫ 2B ෇ 0 2A ϩ B Ϫ 2C ෇ 0 The solution of this system of equations is A ෇ Ϫ1 2 yp+2f+3g yp+3g Ϫ2A ෇ 1 B ෇ Ϫ1 2 C ෇ Ϫ3 4 A particular solution is therefore yp+2f _3 3 yp͑x͒ ෇ Ϫ 1 x 2 Ϫ 1 x Ϫ 3 2 2 4 yp and, by Theorem 3, the general solution is _5 y ෇ yc ϩ yp ෇ c1 e x ϩ c2 eϪ2x Ϫ 1 x 2 Ϫ 1 x Ϫ 3 2 2 4 FIGURE 1 If G͑x͒ (the right side of Equation 1) is of the form Ce k x, where C and k are constants, then we take as a trial solution a function of the same form, yp͑x͒ ෇ Ae k x, because the derivatives of e k x are constant multiples of e k x. EXAMPLE 2 Solve yЉ ϩ 4y ෇ e 3x. SOLUTION The auxiliary equation is r 2 ϩ 4 ෇ 0 with roots Ϯ2i, so the solution of the Figure 2 shows solutions of the differential equation in Example 2 in terms of yp and the functions f ͑x͒ ෇ cos 2x and t͑x͒ ෇ sin 2x. Notice that all solutions approach ϱ as x l ϱ and all solutions resemble sine functions when x is negative. ■ ■ 4 complementary equation is yc͑x͒ ෇ c1 cos 2x ϩ c2 sin 2x For a particular solution we try yp͑x͒ ෇ Ae 3x. Then yp ෇ 3Ae 3x and ypЉ ෇ 9Ae 3x. SubstiЈ tuting into the differential equation, we have 9Ae 3x ϩ 4͑Ae 3x ͒ ෇ e 3x yp+f+g yp+g yp _4 yp+f _2 FIGURE 2 1 so 13Ae 3x ෇ e 3x and A ෇ 13 . Thus, a particular solution is 2 1 yp͑x͒ ෇ 13 e 3x and the general solution is 1 y͑x͒ ෇ c1 cos 2x ϩ c2 sin 2x ϩ 13 e 3x If G͑x͒ is either C cos kx or C sin kx, then, because of the rules for differentiating the sine and cosine functions, we take as a trial particular solution a function of the form yp͑x͒ ෇ A cos kx ϩ B sin kx
• 3. NONHOMOGENEOUS LINEAR EQUATIONS ■ 3 EXAMPLE 3 Solve yЉ ϩ yЈ Ϫ 2y ෇ sin x. SOLUTION We try a particular solution yp͑x͒ ෇ A cos x ϩ B sin x yp ෇ ϪA sin x ϩ B cos x Ј Then ypЉ ෇ ϪA cos x Ϫ B sin x so substitution in the differential equation gives ͑ϪA cos x Ϫ B sin x͒ ϩ ͑ϪA sin x ϩ B cos x͒ Ϫ 2͑A cos x ϩ B sin x͒ ෇ sin x ͑Ϫ3A ϩ B͒ cos x ϩ ͑ϪA Ϫ 3B͒ sin x ෇ sin x or This is true if Ϫ3A ϩ B ෇ 0 and ϪA Ϫ 3B ෇ 1 The solution of this system is 1 A ෇ Ϫ 10 3 B ෇ Ϫ 10 so a particular solution is 1 3 yp͑x͒ ෇ Ϫ 10 cos x Ϫ 10 sin x In Example 1 we determined that the solution of the complementary equation is yc ෇ c1 e x ϩ c2 eϪ2x. Thus, the general solution of the given equation is 1 y͑x͒ ෇ c1 e x ϩ c2 eϪ2x Ϫ 10 ͑cos x ϩ 3 sin x͒ If G͑x͒ is a product of functions of the preceding types, then we take the trial solution to be a product of functions of the same type. For instance, in solving the differential equation yЉ ϩ 2yЈ ϩ 4y ෇ x cos 3x we would try yp͑x͒ ෇ ͑Ax ϩ B͒ cos 3x ϩ ͑Cx ϩ D͒ sin 3x If G͑x͒ is a sum of functions of these types, we use the easily veriﬁed principle of superposition, which says that if yp1 and yp2 are solutions of ayЉ ϩ byЈ ϩ cy ෇ G1͑x͒ ayЉ ϩ byЈ ϩ cy ෇ G2͑x͒ respectively, then yp1 ϩ yp2 is a solution of ayЉ ϩ byЈ ϩ cy ෇ G1͑x͒ ϩ G2͑x͒ EXAMPLE 4 Solve yЉ Ϫ 4y ෇ xe x ϩ cos 2x. SOLUTION The auxiliary equation is r 2 Ϫ 4 ෇ 0 with roots Ϯ2, so the solution of the com- plementary equation is yc͑x͒ ෇ c1 e 2x ϩ c2 eϪ2x. For the equation yЉ Ϫ 4y ෇ xe x we try yp1͑x͒ ෇ ͑Ax ϩ B͒e x Ј Љ Then yp1 ෇ ͑Ax ϩ A ϩ B͒e x, yp1 ෇ ͑Ax ϩ 2A ϩ B͒e x, so substitution in the equation gives ͑Ax ϩ 2A ϩ B͒e x Ϫ 4͑Ax ϩ B͒e x ෇ xe x or ͑Ϫ3Ax ϩ 2A Ϫ 3B͒e x ෇ xe x
• 4. 4 ■ NONHOMOGENEOUS LINEAR EQUATIONS Thus, Ϫ3A ෇ 1 and 2A Ϫ 3B ෇ 0, so A ෇ Ϫ 1 , B ෇ Ϫ 2 , and 3 9 yp1͑x͒ ෇ (Ϫ 1 x Ϫ 2 )e x 3 9 For the equation yЉ Ϫ 4y ෇ cos 2x, we try yp2͑x͒ ෇ C cos 2x ϩ D sin 2x In Figure 3 we show the particular solution yp ෇ yp1 ϩ yp 2 of the differential equation in Example 4. The other solutions are given in terms of f ͑x͒ ෇ e 2 x and t͑x͒ ෇ eϪ2 x. ■ ■ 5 Substitution gives Ϫ4C cos 2x Ϫ 4D sin 2x Ϫ 4͑C cos 2x ϩ D sin 2x͒ ෇ cos 2x Ϫ8C cos 2x Ϫ 8D sin 2x ෇ cos 2x or yp+2f+g Therefore, Ϫ8C ෇ 1, Ϫ8D ෇ 0, and yp+g yp+f _4 yp2͑x͒ ෇ Ϫ 1 cos 2x 8 1 yp By the superposition principle, the general solution is _2 y ෇ yc ϩ yp1 ϩ yp2 ෇ c1 e 2x ϩ c2 eϪ2x Ϫ ( 1 x ϩ 2 )e x Ϫ 1 cos 2x 3 9 8 FIGURE 3 Finally we note that the recommended trial solution yp sometimes turns out to be a solution of the complementary equation and therefore can’t be a solution of the nonhomogeneous equation. In such cases we multiply the recommended trial solution by x (or by x 2 if necessary) so that no term in yp͑x͒ is a solution of the complementary equation. EXAMPLE 5 Solve yЉ ϩ y ෇ sin x. SOLUTION The auxiliary equation is r 2 ϩ 1 ෇ 0 with roots Ϯi, so the solution of the com- plementary equation is yc͑x͒ ෇ c1 cos x ϩ c2 sin x Ordinarily, we would use the trial solution yp͑x͒ ෇ A cos x ϩ B sin x but we observe that it is a solution of the complementary equation, so instead we try yp͑x͒ ෇ Ax cos x ϩ Bx sin x Then yЈ͑x͒ ෇ A cos x Ϫ Ax sin x ϩ B sin x ϩ Bx cos x p ypЉ͑x͒ ෇ Ϫ2A sin x Ϫ Ax cos x ϩ 2B cos x Ϫ Bx sin x The graphs of four solutions of the differential equation in Example 5 are shown in Figure 4. ■ ■ Substitution in the differential equation gives 4 ypЉ ϩ yp ෇ Ϫ2A sin x ϩ 2B cos x ෇ sin x so A ෇ Ϫ 1 , B ෇ 0, and 2 _2π 2π yp͑x͒ ෇ Ϫ 1 x cos x 2 yp _4 FIGURE 4 The general solution is y͑x͒ ෇ c1 cos x ϩ c2 sin x Ϫ 1 x cos x 2
• 5. NONHOMOGENEOUS LINEAR EQUATIONS ■ 5 We summarize the method of undetermined coefﬁcients as follows: 1. If G͑x͒ ෇ e kxP͑x͒, where P is a polynomial of degree n, then try yp͑x͒ ෇ e kxQ͑x͒, where Q͑x͒ is an nth-degree polynomial (whose coefﬁcients are determined by substituting in the differential equation.) 2. If G͑x͒ ෇ e kxP͑x͒ cos mx or G͑x͒ ෇ e kxP͑x͒ sin mx, where P is an nth-degree polynomial, then try yp͑x͒ ෇ e kxQ͑x͒ cos mx ϩ e kxR͑x͒ sin mx where Q and R are nth-degree polynomials. Modiﬁcation: If any term of yp is a solution of the complementary equation, multiply yp by x (or by x 2 if necessary). EXAMPLE 6 Determine the form of the trial solution for the differential equation yЉ Ϫ 4yЈ ϩ 13y ෇ e 2x cos 3x. SOLUTION Here G͑x͒ has the form of part 2 of the summary, where k ෇ 2, m ෇ 3, and P͑x͒ ෇ 1. So, at ﬁrst glance, the form of the trial solution would be yp͑x͒ ෇ e 2x͑A cos 3x ϩ B sin 3x͒ But the auxiliary equation is r 2 Ϫ 4r ϩ 13 ෇ 0, with roots r ෇ 2 Ϯ 3i, so the solution of the complementary equation is yc͑x͒ ෇ e 2x͑c1 cos 3x ϩ c2 sin 3x͒ This means that we have to multiply the suggested trial solution by x. So, instead, we use yp͑x͒ ෇ xe 2x͑A cos 3x ϩ B sin 3x͒ The Method of Variation of Parameters Suppose we have already solved the homogeneous equation ayЉ ϩ byЈ ϩ cy ෇ 0 and written the solution as 4 y͑x͒ ෇ c1 y1͑x͒ ϩ c2 y2͑x͒ where y1 and y2 are linearly independent solutions. Let’s replace the constants (or parameters) c1 and c2 in Equation 4 by arbitrary functions u1͑x͒ and u2͑x͒. We look for a particular solution of the nonhomogeneous equation ayЉ ϩ byЈ ϩ cy ෇ G͑x͒ of the form 5 yp͑x͒ ෇ u1͑x͒y1͑x͒ ϩ u2͑x͒y2͑x͒ (This method is called variation of parameters because we have varied the parameters c1 and c2 to make them functions.) Differentiating Equation 5, we get 6 yp ෇ ͑u1 y1 ϩ u2 y2 ͒ ϩ ͑u1 y1 ϩ u2 y2 ͒ Ј Ј Ј Ј Ј Since u1 and u2 are arbitrary functions, we can impose two conditions on them. One condition is that yp is a solution of the differential equation; we can choose the other condition so as to simplify our calculations. In view of the expression in Equation 6, let’s impose the condition that 7 Ј Ј u1 y1 ϩ u2 y2 ෇ 0
• 6. 6 ■ NONHOMOGENEOUS LINEAR EQUATIONS ypЉ ෇ u1 y1 ϩ u2 y2 ϩ u1 y1Љ ϩ u2 y2Љ Ј Ј Ј Ј Then Substituting in the differential equation, we get a͑u1 y1 ϩ u2 y2 ϩ u1 y1Љ ϩ u2 y2Љ͒ ϩ b͑u1 y1 ϩ u2 y2 ͒ ϩ c͑u1 y1 ϩ u2 y2 ͒ ෇ G Ј Ј Ј Ј Ј Ј or 8 u1͑ay1Љ ϩ by1 ϩ cy1 ͒ ϩ u2͑ay2Љ ϩ by2 ϩ cy2 ͒ ϩ a͑u1 y1 ϩ u2 y2 ͒ ෇ G Ј Ј Ј Ј Ј Ј But y1 and y2 are solutions of the complementary equation, so ay1Љ ϩ by1 ϩ cy1 ෇ 0 Ј and ay2Љ ϩ by2 ϩ cy2 ෇ 0 Ј and Equation 8 simpliﬁes to a͑u1 y1 ϩ u2 y2 ͒ ෇ G Ј Ј Ј Ј 9 Equations 7 and 9 form a system of two equations in the unknown functions uЈ and uЈ . 1 2 After solving this system we may be able to integrate to ﬁnd u1 and u2 and then the particular solution is given by Equation 5. EXAMPLE 7 Solve the equation yЉ ϩ y ෇ tan x, 0 Ͻ x Ͻ ␲ ͞2. SOLUTION The auxiliary equation is r 2 ϩ 1 ෇ 0 with roots Ϯi, so the solution of yЉ ϩ y ෇ 0 is c1 sin x ϩ c2 cos x. Using variation of parameters, we seek a solution of the form yp͑x͒ ෇ u1͑x͒ sin x ϩ u2͑x͒ cos x Then yp ෇ ͑u1 sin x ϩ u2 cos x͒ ϩ ͑u1 cos x Ϫ u2 sin x͒ Ј Ј Ј Set u1 sin x ϩ u2 cos x ෇ 0 Ј Ј 10 Then ypЉ ෇ u1 cos x Ϫ u2 sin x Ϫ u1 sin x Ϫ u2 cos x Ј Ј For yp to be a solution we must have ypЉ ϩ yp ෇ u1 cos x Ϫ u2 sin x ෇ tan x Ј Ј 11 Solving Equations 10 and 11, we get u1͑sin 2x ϩ cos 2x͒ ෇ cos x tan x Ј ■ ■ Figure 5 shows four solutions of the differential equation in Example 7. u1 ෇ sin x Ј 2.5 u1͑x͒ ෇ Ϫcos x (We seek a particular solution, so we don’t need a constant of integration here.) Then, from Equation 10, we obtain 0 yp π 2 u2 ෇ Ϫ Ј sin x sin 2x cos 2x Ϫ 1 u1 ෇ Ϫ Ј ෇ ෇ cos x Ϫ sec x cos x cos x cos x _1 FIGURE 5 So u2͑x͒ ෇ sin x Ϫ ln͑sec x ϩ tan x͒
• 7. NONHOMOGENEOUS LINEAR EQUATIONS ■ 7 (Note that sec x ϩ tan x Ͼ 0 for 0 Ͻ x Ͻ ␲ ͞2.) Therefore yp͑x͒ ෇ Ϫcos x sin x ϩ ͓sin x Ϫ ln͑sec x ϩ tan x͔͒ cos x ෇ Ϫcos x ln͑sec x ϩ tan x͒ and the general solution is y͑x͒ ෇ c1 sin x ϩ c2 cos x Ϫ cos x ln͑sec x ϩ tan x͒ Exercises A Click here for answers. S 16. yЉ ϩ 3yЈ Ϫ 4 y ෇ ͑x 3 ϩ x͒e x Click here for solutions. 17. yЉ ϩ 2 yЈ ϩ 10 y ෇ x 2eϪx cos 3x 18. yЉ ϩ 4y ෇ e 3x ϩ x sin 2x 1–10 Solve the differential equation or initial-value problem using the method of undetermined coefﬁcients. 1. yЉ ϩ 3yЈ ϩ 2y ෇ x 2. yЉ ϩ 9y ෇ e 2 3. yЉ Ϫ 2yЈ ෇ sin 4x 7. yЉ ϩ y ෇ e x ϩ x 3, y͑0͒ ෇ 2, 8. yЉ Ϫ 4y ෇ e x cos x, y͑0͒ ෇ 2, ■ ■ ■ ■ ■ ■ yЈ͑0͒ ෇ 0 ■ ■ ■ ■ ■ ; 11–12 Graph the particular solution and several other solutions. What characteristics do these solutions have in common? ■ ■ ■ ■ 13 –18 ■ 13. yЉ ϩ 9 y ෇ e ϩ x sin x 2 Ϫx 14. yЉ ϩ 9 yЈ ෇ xe cos ␲ x 15. yЉ ϩ 9 yЈ ෇ 1 ϩ xe 9x ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 23–28 Solve the differential equation using the method of variation of parameters. 1 1 ϩ eϪx 25. yЉ Ϫ 3yЈ ϩ 2y ෇ ■ ■ ■ ■ ■ Write a trial solution for the method of undetermined coefﬁcients. Do not determine the coefﬁcients. 2x ■ 24. yЉ ϩ y ෇ cot x, 0 Ͻ x Ͻ ␲͞2 12. 2yЉ ϩ 3yЈ ϩ y ෇ 1 ϩ cos 2x ■ ■ 23. yЉ ϩ y ෇ sec x, 0 Ͻ x Ͻ ␲͞2 11. 4yЉ ϩ 5yЈ ϩ y ෇ e x ■ ■ 22. yЉ Ϫ yЈ ෇ e x y͑0͒ ෇ 1, ■ ■ 21. yЉ Ϫ 2yЈ ϩ y ෇ e 2x yЈ͑0͒ ෇ 2 ■ ■ ■ 20. yЉ Ϫ 3yЈ ϩ 2y ෇ sin x yЈ͑0͒ ෇ 1 10. yЉ ϩ yЈ Ϫ 2y ෇ x ϩ sin 2x, ■ 19. yЉ ϩ 4y ෇ x yЈ͑0͒ ෇ 0 y͑0͒ ෇ 1, ■ Solve the differential equation using (a) undetermined coefﬁcients and (b) variation of parameters. 6. yЉ ϩ 2yЈ ϩ y ෇ xeϪx 5. yЉ Ϫ 4yЈ ϩ 5y ෇ e ■ 19–22 4. yЉ ϩ 6yЈ ϩ 9y ෇ 1 ϩ x Ϫx 9. yЉ Ϫ yЈ ෇ xe x, ■ 3x ■ 26. yЉ ϩ 3yЈ ϩ 2y ෇ sin͑e x ͒ 27. yЉ Ϫ y ෇ 1 x eϪ2x x3 28. yЉ ϩ 4yЈ ϩ 4y ෇ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■
• 8. 8 ■ NONHOMOGENEOUS LINEAR EQUATIONS Answers S Click here for solutions. 1. y ෇ c1 eϪ2x ϩ c2 eϪx ϩ 2 x 2 Ϫ 2 x ϩ 1 3. y ෇ c1 ϩ c2 e 2x ϩ 3 7 4 1 cos 4x Ϫ 20 sin 4x 1 5. y ෇ e ͑c1 cos x ϩ c2 sin x͒ ϩ 10 eϪx 3 11 1 x 7. y ෇ 2 cos x ϩ 2 sin x ϩ 2 e ϩ x 3 Ϫ 6x 1 9. y ෇ e x ( 2 x 2 Ϫ x ϩ 2) 5 11. The solutions are all asymptotic to yp ෇ e x͞10 as x l ϱ. Except for yp , all solutions approach _2 4 yp either ϱ or Ϫϱ as x l Ϫϱ. 1 40 2x _4 13. 15. 17. 19. 21. 23. 25. 27. yp ෇ Ae 2x ϩ ͑Bx 2 ϩ Cx ϩ D͒ cos x ϩ ͑Ex 2 ϩ Fx ϩ G͒ sin x yp ෇ Ax ϩ ͑Bx ϩ C ͒e 9x yp ෇ xeϪx ͓͑Ax 2 ϩ Bx ϩ C ͒ cos 3x ϩ ͑Dx 2 ϩ Ex ϩ F͒ sin 3x͔ y ෇ c1 cos 2x ϩ c2 sin 2x ϩ 1 x 4 y ෇ c1e x ϩ c2 xe x ϩ e 2x y ෇ ͑c1 ϩ x͒ sin x ϩ ͑c2 ϩ ln cos x͒ cos x y ෇ ͓c1 ϩ ln͑1 ϩ eϪx ͔͒e x ϩ ͓c2 Ϫ eϪx ϩ ln͑1 ϩ eϪx ͔͒e 2x y ෇ [c1 Ϫ 1 x ͑e x͞x͒ dx]eϪx ϩ [c2 ϩ 1 x ͑eϪx͞x͒ dx]e x 2 2
• 9. NONHOMOGENEOUS LINEAR EQUATIONS ■ 9 Solutions: Nonhomogeneous Linear Equations 1. The auxiliary equation is r 2 + 3r + 2 = (r + 2)(r + 1) = 0, so the complementary solution is 0 yc (x) = c1 e−2x + c2 e−x . We try the particular solution yp (x) = Ax2 + Bx + C, so yp = 2Ax + B and 00 yp = 2A. Substituting into the differential equation, we have (2A) + 3(2Ax + B) + 2(Ax2 + Bx + C) = x2 or 2Ax2 + (6A + 2B)x + (2A + 3B + 2C) = x2 . Comparing coefﬁcients gives 2A = 1, 6A + 2B = 0, and 2A + 3B + 2C = 0, so A = 1 , B = − 3 , and C = 2 2 −2x y(x) = yc (x) + yp (x) = c1 e + c2 e −x + 1 2 x 2 − 7 . 4 3 x 2 Thus the general solution is + 7. 4 3. The auxiliary equation is r 2 − 2r = r(r − 2) = 0, so the complementary solution is yc (x) = c1 + c2 e2x . Try the 0 particular solution yp (x) = A cos 4x + B sin 4x, so yp = −4A sin 4x + 4B cos 4x 00 and yp = −16A cos 4x − 16B sin 4x. Substitution into the differential equation gives (−16A cos 4x − 16B sin 4x) − 2(−4A sin 4x + 4B cos 4x) = sin 4x ⇒ (−16A − 8B) cos 4x + (8A − 16B) sin 4x = sin 4x. Then −16A − 8B = 0 and 8A − 16B = 1 1 and B = − 20 . Thus the general solution is y(x) = yc (x) + yp (x) = c1 + c2 e2x + 1 40 cos 4x − 1 20 ⇒ A= 1 40 sin 4x. 5. The auxiliary equation is r 2 − 4r + 5 = 0 with roots r = 2 ± i, so the complementary solution is 0 00 yc (x) = e2x (c1 cos x + c2 sin x). Try yp (x) = Ae−x , so yp = −Ae−x and yp = Ae−x . Substitution gives Ae−x − 4(−Ae−x ) + 5(Ae−x ) = e−x 2x y(x) = e (c1 cos x + c2 sin x) + ⇒ 10Ae−x = e−x ⇒ A= 1 . 10 Thus the general solution is 1 −x e . 10 7. The auxiliary equation is r 2 + 1 = 0 with roots r = ±i, so the complementary solution is 0 00 yc (x) = c1 cos x + c2 sin x. For y00 + y = ex try yp1 (x) = Aex . Then yp1 = yp1 = Aex and substitution gives Aex + Aex = ex A = 1 , so yp1 (x) = 1 ex . For y00 + y = x3 try yp2 (x) = Ax3 + Bx2 + Cx + D. 2 2 ⇒ 0 00 Then yp2 = 3Ax2 + 2Bx + C and yp2 = 6Ax + 2B. Substituting, we have 6Ax + 2B + Ax3 + Bx2 + Cx + D = x3 , so A = 1, B = 0, 6A + C = 0 ⇒ C = −6, and 2B + D = 0 ⇒ D = 0. Thus yp2 (x) = x3 − 6x and the general solution is y(x) = yc (x) + yp1 (x) + yp2 (x) = c1 cos x + c2 sin x + 1 ex + x3 − 6x. But 2 = y(0) = c1 + 2 and 0 = y0 (0) = c2 + y(x) = 3 2 cos x + 11 2 1 2 −6 ⇒ c2 = 11 . 2 1 2 ⇒ c1 = 3 2 Thus the solution to the initial-value problem is sin x + 1 ex + x3 − 6x. 2 9. The auxiliary equation is r 2 − r = 0 with roots r = 0, r = 1 so the complementary solution is yc (x) = c1 + c2 ex . Try yp (x) = x(Ax + B)ex so that no term in yp is a solution of the complementary equation. Then 0 00 yp = (Ax2 + (2A + B)x + B)ex and yp = (Ax2 + (4A + B)x + (2A + 2B))ex . Substitution into the differential equation gives (Ax2 + (4A + B)x + (2A + 2B))ex − (Ax2 + (2A + B)x + B)ex = xex ⇒ ¡ ¢ (2Ax + (2A + B))ex = xex ⇒ A = 1 , B = −1. Thus yp (x) = 1 x2 − x ex and the general solution is 2 2 ¡ ¢ y(x) = c1 + c2 ex + 1 x2 − x ex . But 2 = y(0) = c1 + c2 and 1 = y0 (0) = c2 − 1, so c2 = 2 and c1 = 0. The 2 ¡ ¡ ¢ ¢ solution to the initial-value problem is y(x) = 2ex + 1 x2 − x ex = ex 1 x2 − x + 2 . 2 2
• 10. 10 ■ NONHOMOGENEOUS LINEAR EQUATIONS 11. yc (x) = c1 e−x/4 + c2 e−x . Try yp (x) = Aex . Then 10Aex = ex , so A = 1 10 and the general solution is y(x) = c1 e−x/4 + c2 e−x + 1 x e . 10 The solutions are all composed of exponential curves and with the exception of the particular solution (which approaches 0 as x → −∞), they all approach either ∞ or −∞ as x → −∞. As x → ∞, all solutions are asymptotic to yp = 1 x e . 10 13. Here yc (x) = c1 cos 3x + c2 sin 3x. For y00 + 9y = e2x try yp1 (x) = Ae2x and for y 00 + 9y = x2 sin x try yp2 (x) = (Bx2 + Cx + D) cos x + (Ex2 + F x + G) sin x. Thus a trial solution is yp (x) = yp1 (x) + yp2 (x) = Ae2x + (Bx2 + Cx + D) cos x + (Ex2 + F x + G) sin x. 15. Here yc (x) = c1 + c2 e−9x . For y 00 + 9y0 = 1 try yp1 (x) = Ax (since y = A is a solution to the complementary equation) and for y 00 + 9y0 = xe9x try yp2 (x) = (Bx + C)e9x . 17. Since yc (x) = e−x (c1 cos 3x + c2 sin 3x) we try yp (x) = x(Ax2 + Bx + C)e−x cos 3x + x(Dx2 + Ex + F )e−x sin 3x (so that no term of yp is a solution of the complementary equation). Note: Solving Equations (7) and (9) in The Method of Variation of Parameters gives u0 = − 1 Gy2 0 0 a (y1 y2 − y2 y1 ) u0 = 2 and Gy1 0 0 a (y1 y2 − y2 y1 ) We will use these equations rather than resolving the system in each of the remaining exercises in this section. 19. (a) The complementary solution is yc (x) = c1 cos 2x + c2 sin 2x. A particular solution is of the form yp (x) = Ax + B. Thus, 4Ax + 4B = x ⇒ A= 1 4 solution is y = yc + yp = c1 cos 2x + c2 sin 2x + 1 x. 4 and B = 0 ⇒ yp (x) = 1 x. Thus, the general 4 (b) In (a), yc (x) = c1 cos 2x + c2 sin 2x, so set y1 = cos 2x, y2 = sin 2x. Then 0 0 y1 y2 − y2 y1 = 2 cos2 2x + 2 sin2 2x = 2 so u0 = − 1 x sin 2x ⇒ 1 2 R ¡ ¢ u1 (x) = − 1 x sin 2x dx = − 1 −x cos 2x + 1 sin 2x [by parts] and u0 = 1 x cos 2x 2 2 4 2 2 R ¡ ¢ ⇒ u2 (x) = 1 x cos 2xdx = 1 x sin 2x + 1 cos 2x [by parts]. Hence 4 2 ¢ ¢ ¡2 ¡ yp (x) = − 1 −x cos 2x + 1 sin 2x cos 2x + 1 x sin 2x + 1 cos 2x sin 2x = 1 x. Thus 4 2 4 2 4 y(x) = yc (x) + yp (x) = c1 cos 2x + c2 sin 2x + 1 x. 4 21. (a) r 2 − r = r(r − 1) = 0 ⇒ r = 0, 1, so the complementary solution is yc (x) = c1 ex + c2 xex . A particular solution is of the form yp (x) = Ae2x . Thus 4Ae2x − 4Ae2x + Ae2x = e2x ⇒ 2x x ⇒ Ae2x = e2x x yp (x) = e . So a general solution is y(x) = yc (x) + yp (x) = c1 e + c2 xe + e2x . ⇒ A=1 0 0 (b) From (a), yc (x) = c1 ex + c2 xex , so set y1 = ex , y2 = xex . Then, y1 y2 − y2 y1 = e2x (1 + x) − xe2x = e2x R x 0 x x and so u1 = −xe ⇒ u1 (x) = − xe dx = −(x − 1)e [by parts] and u0 = ex ⇒ 2 R x u2 (x) = e dx = ex . Hence yp (x) = (1 − x)e2x + xe2x = e2x and the general solution is y(x) = yc (x) + yp (x) = c1 ex + c2 xex + e2x .
• 11. NONHOMOGENEOUS LINEAR EQUATIONS ■ 11 23. As in Example 6, yc (x) = c1 sin x + c2 cos x, so set y1 = sin x, y2 = cos x. Then sec x cos x 0 0 y1 y2 − y2 y1 = − sin2 x − cos2 x = −1, so u0 = − = 1 ⇒ u1 (x) = x and 1 −1 R sec x sin x u0 = = − tan x ⇒ u2 (x) = − tan xdx = ln |cos x| = ln(cos x) on 0 < x < π . Hence 2 2 −1 yp (x) = x sin x + cos x ln(cos x) and the general solution is y(x) = (c1 + x) sin x + [c2 + ln(cos x)] cos x. 0 0 25. y1 = ex , y2 = e2x and y1 y2 − y2 y1 = e3x . So u0 = 1 Z −e2x e−x =− and −x )e3x (1 + e 1 + e−x e−x ex ex dx = ln(1 + e−x ). u0 = = 3x so 2 1 + e−x (1 + e−x )e3x e + e2x ¶ µ x Z ex e +1 u2 (x) = − e−x = ln(1 + e−x ) − e−x . Hence dx = ln 3x + e2x e ex u1 (x) = − yp (x) = ex ln(1 + e−x ) + e2x [ln(1 + e−x ) − e−x ] and the general solution is y(x) = [c1 + ln(1 + e−x )]ex + [c2 − e−x + ln(1 + e−x )]e2x . ex e−x 0 0 and 27. y1 = e−x , y2 = ex and y1 y2 − y2 y1 = 2. So u0 = − , u0 = 1 2 2x 2x Z x Z −x e e dx + ex dx. Hence the general solution is yp (x) = −e−x 2x 2x µ µ ¶ Z x Z −x ¶ e e y(x) = c1 − dx e−x + c2 + dx ex . 2x 2x