respect to x. ng sides of Supplementary Problems Use variation of parameters to find the general solutions of the following differential equations: 12.9.) y" - 2y' + y = et 12.10. y" +y= sec x 12.11. y" -y'- 2y = e3* 12.12. y" – 60y' – 900y = 5e10x %3D 12.13.) y"- 7y' =- 3 12.14. y"+y- y = In x if two solutions to the associated homogeneous problem are known to be x and 17x. 12.15. xy"- xy' x'e* if two solutions to the associated homogeneous problem are known to be 1 and x". (1) 12.16. y--y=x² and v =x19. y +2xy = x 12.17. %3D e' * - 2x + x =- 12.18. y" = 12 %3D 12.19. 12.20. й — 6х +9х: *+ 4x = 4 sec? 2t 12.21. e' * - 4x + 3x = 12.22. %3D 1+e' (t – 1)ä – 2tx + 2x = (t – 1) if two solutions to the associated homogeneous equations are known to be t and + 1. 12.23. %3D (t2 +t)ï + (2 – t²)x – (2 +t)x =t(t+1)² if two solutions to the associated homogeneous equations are known to be 12.24. e' and 1/t. (1) e' 12.25.) - 3ii + 3r -r =- ř + 6ř +12r + 8r = 12e2 12.26. %3D d'z 3 e30 dz +2- de 1+ e° 12.28. - 52 + 25ż – 125z = 1000 %3D 12.27. d0² do3 12.29. t°ÿ + 3t²j = 1 if three linearly independent solutions to the associated homogeneous equations are known to be 1/t, 1, and t. y(5) – 4y(3) = 32e2r 12.30. %3D /2

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.