should memorize should know the fiTS ples of reduction of order. (ii) Reduction of order can be used to find the general solution of a nor geneous equation az(x)y" + a(x)y' + ao(x)y = g(x) whenever a...

y''-3y'+2y=5e^3x,y1=e^x

Extracted text: should memorize should know the fiTS ples of reduction of order. (ii) Reduction of order can be used to find the general solution of a nor geneous equation az(x)y" + a(x)y' + ao(x)y = g(x) whenever a solutie the associated homogeneous equation is known. See Problems 17 Exercises 4.2. EXERCISES 4.2 Answers to selected odd-nmbered problems begin on In Problems 1-16 the indicated function yı(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution yz(x). 7. 9y" - 12y' + 4y = 0; yı = e24/3 8. 6y" + y' - y = 0; yı = e 9. xy" - 7xy' + 16y = 0; y1 = x* 10. xy" + 2xy' - 6y = 0; y1 = x 11. xy" + y' = 0; yı = In x 1. y" - 4y' + 4y = 0; y = * 2. y" + 2y' + y = 0; yı = xe* 3. y" + 16y = 0; yı = cos 4x 4. y" + 9y = 0; y = sin 3x 12. 4x*y" + y = 0; y = x2 In x 13. xy" - xy' + 2y = 0; yı = x sin(In x) 5. y" - y = 0; y = cosh x 6. y - 25y = 0; y = 14. xy" - 3ry' + 5y = 0; yı = x cos(ln x) 4.3 HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS y2 = e":* ar of the form y, = xe"", m 15. (1 - 2x - x)y" + 2(1 + x)y' - 2y = 0; yı =x + 1 16. (1 - x)y" + 2.xy' = 0; 1 = 1 constants. (c) Reexamine Problems 1-8. Can you expla statements in parts (a) and (b) above contradicted by the answers to Problems In Problems 17-20 the indicated function y,(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution yx) of the homogeneous equation and a particular solution of the given nonhomogeneous equation. 22. Verify that y,(x) = x is a solution of xy" – xy Use reduction of order to find a second solutic the fom of an infinite series. Conjecture an definition for yax). 17. y" - 4y = 2: yı = e2* 18. y" + y = 1; y = 1 Computer Lab Assignments 19. y" - 3y' + 2y = 5e: y1 = e* 23. (a) Verify that yı(x) = e* is a solution of 20. y" - 4y' + 3y = x; y1 = e* ху" - (х + 10)у' + 10у0. Discussion Problems (b) Use (5) to find a second solution y(x). Usa carry out the required integration. (c) Explain, using Coollary (A) of Theorem - the second solution can be written compac 21. (a) Give a convincing demonstration that the second- crder equation ay" + by' + cy = 0, a, b, and e con- stants, always possesses at least one solution of the brm y = e", m, a constant. 10 1 (b) Explain why the differential equation in part (a) must then have a second solution either of the form yx) = 4.3 HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS REVIEW MATERIAL • Review Problem 27 in Exercises 1.1 and Theorem 4.1.5 • Review the algebra of solving polynomial equations (see the Student Resource and Solutions Manual) INTRODUCTION As a means of motivating the discussion in this section, let us return to order differential equations-more specifically, to homogeneous linear equations ay' + by where the coefficients a + 0 and b are constants. This type of equation can be solved eithe separation of variables or with the aid of an integrating factor, but there is another solution met anlu algabra Rafora illutrating this altarnati