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Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations - SECTION 17.6...

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SECTION 17.6: Nonhomogeneous Linear Equations 923 The corresponding powers x r can be expressed in real form in a manner similar to that used for constant coefficient equations; we have x α ± i β = e ± i β) ln x = e α ln x cos ln x ) ± i sin ln x ) = x α cos ln x ) ± ix α sin ln x ). Accordingly, the Euler equation has the general solution y = C 1 | x | α cos ln | x | ) + C 2 | x | α sin ln | x | ). Example 4 Solve the DE x 2 y 3 xy + 13 y = 0. Solution The DE has the auxiliary equation r ( r 1 ) 3 r + 13 = 0, that is, r 2 4 r + 13 = 0, which has roots r = 2 ± 3 i . The DE, therefore, has the general solution y = C 1 x 2 cos ( 3 ln | x | ) + C 2 x 2 sin ( 3 ln | x | ). Exercises 17.5 Exercises involving the solution of second-order, linear, homogeneous equations with constant coefficients can be found at the end of Section 3.7. Find general solutions of the DEs in Exercises 1–4. 1. y 4 y + 3 y = 0 2. y ( 4 ) 2 y + y = 0 3. y ( 4 ) + 2 y + y = 0 4. y ( 4 ) + 4 y ( 3 ) + 6 y + 4 y + y = 0 5. Show that y = e 2 t is a solution of y 2 y 4 y = 0 (where denotes d / dt ), and find the general solution of this DE. 6. Write the general solution of the linear, constant-coefficient DE having auxiliary equation ( r 2 r 2 ) 2 ( r 2 4 ) 2 = 0. Find general solutions to the Euler equations in Exercises 7–12. 7. x 2 y xy + y = 0 8. x 2 y xy 3 y = 0 9. x 2 y + xy y = 0 10. x 2 y xy + 5 y = 0 11. x 2 y + xy = 0 12. x 2 y + xy + y = 0 13. 3 Solve the DE x 3 y + xy y = 0 in the interval x > 0. 17.6 Nonhomogeneous Linear Equations We now consider the problem of solving the nonhomogeneoussecond-orderdifferential equation a 2 ( x ) d 2 y dx 2 + a 1 ( x ) dy dx + a 0 ( x ) y = f ( x ). ( ) We assume that two independent solutions, y 1 ( x ) and y 2 ( x ) , of the corresponding homogeneous equation a 2 ( x ) d 2 y dx 2 + a 1 ( x ) dy dx + a 0 ( x ) y = 0 are known. The function y h ( x ) = C 1 y 1 ( x ) + C 2 y 2 ( x ) , which is the general solution of the homogeneous equation, is called the complementary function for the nonho- mogeneous equation. Theorem 2 of Section 17.1 suggests that the general solution of the nonhomogeneous equation is of the form y = y p ( x ) + y h ( x ) = y p ( x ) + C 1 y 1 ( x ) + C 2 y 2 ( x ),
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924 CHAPTER 17 Ordinary Differential Equations where y p ( x ) is any particular solution of the nonhomogeneous equation. All we need to do is find one solution of the nonhomogeneous equation, and we can write the general solution. There are two common methods for finding a particular solution y p of the nonho- mogeneous equation ( ) : 1. the method of undetermined coefficients and 2. the method of variation of parameters. The first of these hardly warrants being called a method ; it just involves making an educated guess about the form of the solution as a sum of terms with unknown coefficients and substituting this guess into the equation to determine the coefficients. This method works well for simple DEs, especially ones with constant coefficients.
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