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 coefFcient equations; we have x α ± i β = e ± i β) ln x = e α ln x ± cos ln x ) ± i sin ln x ) ² = 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 00 3 xy 0 + 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 ( 3ln | x | ) + C 2 x 2 sin ( | x | ). Exercises 17.5 Exercises involving the solution of second-order, linear, homogeneous equations with constant coefFcients can be found at the end of Section 3.7. ±ind general solutions of the DEs in Exercises 1–4. 1. y 000 4 y 00 + 3 y 0 = 0 2. y ( 4 ) 2 y 00 + y = 0 3. y ( 4 ) + 2 y 00 + y = 0 4. y ( 4 ) + 4 y ( 3 ) + 6 y 00 + 4 y 0 + y = 0 5. Show that y = e 2 t is a solution of y 000 2 y 0 4 y = 0 (where 0 denotes d / dt ), and Fnd the general solution of this DE. 6. Write the general solution of the linear, constant-coefFcient DE having auxiliary equation ( r 2 r 2 ) 2 ( r 2 4 ) 2 = 0. ±ind general solutions to the Euler equations in Exercises 7–12. 7. x 2 y 00 0 + y = 0 8. x 2 y 00 0 3 y = 0 9. x 2 y 00 + 0 y = 0 10. x 2 y 00 0 + 5 y = 0 11. x 2 y 00 + 0 = 0 12. x 2 y 00 + 0 + y = 0 13. 3 Solve the DE x 3 y 000 + 0 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 + 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 2 + a 1 ( x ) + 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 Fnd one solution of the nonhomogeneous equation, and we can write the general solution. There are two common methods for Fnding a particular solution y p of the nonho- mogeneous equation ( ) : 1. the method of undetermined coefFcients and 2. the method of variation of parameters. The Frst 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 coefFcients and substituting this guess into the equation to determine the coefFcients. This method works well for simple DEs, especially ones with constant coefFcients.
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This note was uploaded on 03/21/2012 for the course STAT 101 taught by Professor Graham during the Spring '08 term at Iowa State.

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Nonhomogeneous Linear Equations - SECTION 17.6:...

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