y''-3y'+2y=5e^3x,y1=e^x should memorizeshould know the fiTSples of reduction of order.(ii) Reduction of order can be used to find the general solution of a norgeneous equation az(x)y" + a(x)y' + ao(x)y = g(x) whenever a solutiethe associated homogeneous equation is known. See Problems 17Exercises 4.2.EXERCISES 4.2Answers to selected odd-nmbered problems begin onIn Problems 1-16 the indicated function yı(x) is a solutionof the given differential equation. Use reduction of order orformula (5), as instructed, to find a second solution yz(x).7. 9y" - 12y' + 4y = 0; yı = e24/38. 6y" + y' - y = 0; yı = e9. xy" - 7xy' + 16y = 0; y1 = x*10. xy" + 2xy' - 6y = 0; y1 = x11. xy" + y' = 0; yı = In x1. y" - 4y' + 4y = 0; y = *2. y" + 2y' + y = 0; yı = xe*3. y" + 16y = 0; yı = cos 4x4. y" + 9y = 0; y = sin 3x12. 4x*y" + y = 0; y = x2 In x13. xy" - xy' + 2y = 0; yı = x sin(In x)5. y" - y = 0; y = cosh x6. y - 25y = 0; y =14. xy" - 3ry' + 5y = 0; yı = x cos(ln x)4.3 HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTSy2 = e":* ar of the form y, = xe"", m15. (1 - 2x - x)y" + 2(1 + x)y' - 2y = 0; yı =x + 116. (1 - x)y" + 2.xy' = 0; 1 = 1constants.(c) Reexamine Problems 1-8. Can you explastatements in parts (a) and (b) abovecontradicted by the answers to ProblemsIn Problems 17-20 the indicated function y,(x) is a solutionof the associated homogeneous equation. Use the methodof reduction of order to find a second solution yx) of thehomogeneous equation and a particular solution of the givennonhomogeneous equation.22. Verify that y,(x) = x is a solution of xy" – xyUse reduction of order to find a second soluticthe fom of an infinite series. Conjecture andefinition for yax).17. y" - 4y = 2: yı = e2*18. y" + y = 1; y = 1Computer Lab Assignments19. y" - 3y' + 2y = 5e: y1 = e*23. (a) Verify that yı(x) = e* is a solution of20. y" - 4y' + 3y = x; y1 = e*ху" - (х + 10)у' + 10у0.Discussion Problems(b) Use (5) to find a second solution y(x). Usacarry out the required integration.(c) Explain, using Coollary (A) of Theorem -the second solution can be written compac21. (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 thebrm 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 formyx) =4.3HOMOGENEOUS LINEAR EQUATIONSWITH CONSTANT COEFFICIENTSREVIEW MATERIAL• Review Problem 27 in Exercises 1.1 and Theorem 4.1.5• Review the algebra of solving polynomial equations (see the Student Resourceand Solutions Manual)INTRODUCTION As a means of motivating the discussion in this section, let us return toorder differential equations-more specifically, to homogeneous linear equations ay' + bywhere the coefficients a + 0 and b are constants. This type of equation can be solved eitheseparation of variables or with the aid of an integrating factor, but there is another solution metanlu algabra Rafora illutrating this altarnati

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y''-3y'+2y=5e^3x,y1=e^x should memorizeshould know the fiTSples of reduction of order.(ii) Reduction of order can be used to find the general solution of a norgeneous equation az(x)y&#34; + a(x)y' + ao(x)y = g(x) whenever a solutiethe associated homogeneous equation is known. See Problems 17Exercises 4.2.EXERCISES 4.2Answers to selected odd-nmbered problems begin onIn Problems 1-16 the indicated function yı(x) is a solutionof the given differential equation. Use reduction of order orformula (5), as instructed, to find a second solution yz(x).7. 9y&#34; - 12y' + 4y = 0; yı = e24/38. 6y&#34; + y' - y = 0; yı = e9. xy&#34; - 7xy' + 16y = 0; y1 = x*10. xy&#34; + 2xy' - 6y = 0; y1 = x11. xy&#34; + y' = 0; yı = In x1. y&#34; - 4y' + 4y = 0; y = *2. y&#34; + 2y' + y = 0; yı = xe*3. y&#34; + 16y = 0; yı = cos 4x4. y&#34; + 9y = 0; y = sin 3x12. 4x*y&#34; + y = 0; y = x2 In x13. xy&#34; - xy' + 2y = 0; yı = x sin(In x)5. y&#34; - y = 0; y = cosh x6. y - 25y = 0; y =14. xy&#34; - 3ry' + 5y = 0; yı = x cos(ln x)4.3 HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTSy2 = e&#34;:* ar of the form y, = xe&#34;&#34;, m15. (1 - 2x - x)y&#34; + 2(1 + x)y' - 2y = 0; yı =x + 116. (1 - x)y&#34; + 2.xy' = 0; 1 = 1constants.(c) Reexamine Problems 1-8. Can you explastatements in parts (a) and (b) abovecontradicted by the answers to ProblemsIn Problems 17-20 the indicated function y,(x) is a solutionof the associated homogeneous equation. Use the methodof reduction of order to find a second solution yx) of thehomogeneous equation and a particular solution of the givennonhomogeneous equation.22. Verify that y,(x) = x is a solution of xy&#34; – xyUse reduction of order to find a second soluticthe fom of an infinite series. Conjecture andefinition for yax).17. y&#34; - 4y = 2: yı = e2*18. y&#34; + y = 1; y = 1Computer Lab Assignments19. y&#34; - 3y' + 2y = 5e: y1 = e*23. (a) Verify that yı(x) = e* is a solution of20. y&#34; - 4y' + 3y = x; y1 = e*ху&#34; - (х + 10)у' + 10у0.Discussion Problems(b) Use (5) to find a second solution y(x). Usacarry out the required integration.(c) Explain, using Coollary (A) of Theorem -the second solution can be written compac21. (a) Give a convincing demonstration that the second-crder equation ay&#34; + by' + cy = 0, a, b, and e con-stants, always possesses at least one solution of thebrm y = e&#34;, m, a constant.10 1(b) Explain why the differential equation in part (a)must then have a second solution either of the formyx) =4.3HOMOGENEOUS LINEAR EQUATIONSWITH CONSTANT COEFFICIENTSREVIEW MATERIAL• Review Problem 27 in Exercises 1.1 and Theorem 4.1.5• Review the algebra of solving polynomial equations (see the Student Resourceand Solutions Manual)INTRODUCTION As a means of motivating the discussion in this section, let us return toorder differential equations-more specifically, to homogeneous linear equations ay' + bywhere the coefficients a + 0 and b are constants. This type of equation can be solved eitheseparation of variables or with the aid of an integrating factor, but there is another solution metanlu algabra Rafora illutrating this altarnati

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Reduction of order is a very significant and basic method to solve higher-order linear differential…