In each of problems 1-12 Solve the given differential equations and the given IVPS. 1. y (v) +y"=x 2. 16y ) 8y"+ y = 0 (v) 3. (iv) - +3y "- y" 0 (iv) 4. y =x+e* 6. y"-3y"+3y'-y =-3 sin 5x -3x 7. y"+y'=-2 sin 2x y"+3y"+3y'+y = 30 cosx 9. 4y " +8y"+4ly'+37y = 0, y (0)= 9, y'(0)=-6.5, y"(0)=-39.75 (iv) 10. y) - 16y =128 cosh 2x , y (0) = 1, y'(0)= 24, y" (0)=20, y"(0) =- %3D %3D %3D (iv) 11. y +6y "+17y"+22y'+14y = 0, y (0) = 1, y'(0)=-2, y"(0)=0, y"(0)=3 (iv) "+ y"+8y'-12y =12sinx -e, y (0)= 3, y'(0)= 0, y" (0) =-1, y"(0)= 2 In each of problems 13-14 Determine whether the given set of functions in linearly 12. y ) +2y independent or linearly dependent. 13. fi (x ) =x' +1, f;(x)=x +x +1, f; (x)=2x -2 14. f, (x )=e*,f2(x )=e*,f:(x )=1 In each of problems 15-17 Solve the given Euler-Cauchy equations. 15. x 'y "+7x 'y"-2xy'-10y = 0, y (1)=1, y'(1)=-7, y"(1) = 44 3 16. " +*"- 2' +2 %3D2 4 17. * " + *"- '+6 %3D4 ? In each of problems 17-18 Using the method of Variation of Parameters to solve the following equation. 18. y "- y'= x 19. x 'y "+x'y"-2xy'+2y 2x* !! 20. Solve the non-linear equation 2x *(y ")'y" +4x (y")' =y". of