Question

Can you please show number #25, #27 (Please make work readable)

21. y" + 3y" + 3y' + y = 0 22. y" – 6y" + 12y' – 8y = 0 23. y(a) + y + y" =0 24. y(4) – 2y" +y=0 In Problems 1-14 find the general solution of the given second-order differential equation. 1. 4y" + y' = 0 2. y" – 36y = 0 3. y" - y' - 6y = 0 4. y" – 3y' + 2y = 0 5. y" + 8y' + 16y=0 6. y" – 10y' + 25y = 0 7. 12y" – 5y' – 2y = 0 8. y" + 4y' - y=0 9. y" +9y = 0 10. 3y + y=0 11. y" – 4y' + 5y = 0 12. 2y" + 2y' +y=0 13. 3y" + 2y' + y=0 14. 2y" – 3y + 4y = 0 25. 16 26. za +99 = 0 - 189=0 28. 2 - 12 months ago In Problems 15-28 find the general solution of the given higher- order differential equation. 15. y" - 4y" - 5y' = 0 16. y" - y=0 17. y." – 5y" + 3y' +9y = 0 + 3y" – 4y' – 12y = 0 In Problems 29-36 solve the given initial-value problem. 29. y" + 16y = 0, y(0) = 2, y'(O)= -2 30.42 + y = 0, 7(7/3) = 0, y'(7/3) = 2 19. - 2 = 0 20. i - - 48 = 0 31.42 - 4x - 5y = 0, y(i) = 0, y'(1) = 2 32. 4y" – 4y' – 3y = 0, y(0) = 1, y'(0) = 5

Answer 1

We will find the characteristic equation and its roots. Then we find the general solution using these roots. The detailed solution is attached below.

25 The given differential equation is - 16.04 + 24 d y 7 + 99 = 0 The characteristic equation is 16m² + 24 m² +9=0 (4 m+ 3) = 0 * 4m²+3=0 I = m = tij3 (twice) s The solution is y(x) = (@+ (2x) cos 3x + (63+ Cyx) sin JB x where a, C2, C3 and Cu are arbitrary constante 27 The given differenti al equation is C et elementum lorem et 54 -6 The characteristic equation is m3 + 5m" -2m-10m² + m +5=0

= (x+1)} (x-1)} (x+5) = 0 x=-5 ou X = -1 (twice) or x=1 (twice) : The solution is y(x) = Geox + (6 + GXDē" + (cy + GgXJL* where a, 6, C3, Cu and Co are arbitrary constants