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Jo In each of Problems 11 through 13, determine the values of r for which the given differential equation has solutions of the form y = e't. 11. y' + 2y = 0 12. 13. y""-3y" + 2y' = 0 y"+y'-6y=0

Jo In each of Problems 11 through 13, determine the values of r for which the given differential equation has solutions of the form y = e't. 11. y' + 2y = 0 12. 13. y""-3y" + 2y' = 0 y"+y'-6y=0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 49E
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Problems In each of Problems 1 through 4, determine the order of the given differential equation; also state whether the equation is linear or nonlinear. 1. 12d²y 2. dy +1 dt² dt 3. + 2y = sin t (1+ y²) dy d²y dt² dt +t- +y=e¹ dªy d³y d²y dy + + + +y=1 dt4 dt³ dt² dt d²y 4. dt² In each of Problems 5 through 10, verify that each given function is a solution of the differential equation. 5. y" -y=0; y(t) = e, y2(t) = cosht 6. y" +2y' - 3y = 0; y₁(t) = e-³1, y₂(t) = et 7. ty' - y = t²; y = 3t+t² 8. y + 4y+3y= t; y₁(t) = t/3, y2(t) = e+t/3 9. 1²y" +5ty' +4y = 0, t> 0; yi(t) = t2, y₂(t) = t-² Int 10. y' - 2¹y = 1; y=e³² e³ds + e² + sin(t + y) = sint In each of Problems 11 through 13, determine the values of r for which the given differential equation has solutions of the form yet. 11. y' +2y=0 12. y"+y'-6y=0 13. y" - 3y" + 2y' = 0 In each of Problems 14 and 15, determine the values of r for which the given differential equation has solutions of the form y = t' for t > 0. 14. ty" +4ty' + 2y = 0 15. 1²y" - 4ty' + 4y = 0 In each of Problems 16 through 18, determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. 16. Uxx + Uyy + Uzz = 0 17. Uxxxx +2uxxyy + Uyyyy = 0 18. ut + uux = 1 + Uxx 10. In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation. 19. Uxx + Uyy = 0; u₁(x, y) = cos x cosh y, u₂(x, y) = ln(x² + y²) 20. a²uxx = ui; u₂(x, t) = e-α²x²t u₁(x, t) = e-a²t sinx, sin λx, À a real constant 21. a²uxx = Utti u₂(x, t) = sin(x-at), λ a real constant u₁(x, t) = sin(x) sin(at), 22. Follow the steps indicated here to derive the equation of motion of a pendulum, equation (12) in the text. Assume that the rod is rigid and weightless, that the mass is a point mass, and that there is no friction or drag anywhere in the system. to a. Assume that the mass is in an arbitrary displaced position, indicated by the angle 0. Draw a free-body diagram showing the forces acting on the mass. b. Apply Newton's law of motion in the direction tangential to the circular arc on which the mass moves. Then the tensile force in the rod does not enter the equation. Observe that you need to find the component of the gravitational force in the tangential direction. Observe also that the linear acceleration, as opposed to the angular acceleration, is Ld²0/dt², where L is the length of the rod. c. Simplify the result from part b to obtain equation (12) in the text.
Transcribed Image Text:Problems In each of Problems 1 through 4, determine the order of the given differential equation; also state whether the equation is linear or nonlinear. 1. 12d²y 2. dy +1 dt² dt 3. + 2y = sin t (1+ y²) dy d²y dt² dt +t- +y=e¹ dªy d³y d²y dy + + + +y=1 dt4 dt³ dt² dt d²y 4. dt² In each of Problems 5 through 10, verify that each given function is a solution of the differential equation. 5. y" -y=0; y(t) = e, y2(t) = cosht 6. y" +2y' - 3y = 0; y₁(t) = e-³1, y₂(t) = et 7. ty' - y = t²; y = 3t+t² 8. y + 4y+3y= t; y₁(t) = t/3, y2(t) = e+t/3 9. 1²y" +5ty' +4y = 0, t> 0; yi(t) = t2, y₂(t) = t-² Int 10. y' - 2¹y = 1; y=e³² e³ds + e² + sin(t + y) = sint In each of Problems 11 through 13, determine the values of r for which the given differential equation has solutions of the form yet. 11. y' +2y=0 12. y"+y'-6y=0 13. y" - 3y" + 2y' = 0 In each of Problems 14 and 15, determine the values of r for which the given differential equation has solutions of the form y = t' for t > 0. 14. ty" +4ty' + 2y = 0 15. 1²y" - 4ty' + 4y = 0 In each of Problems 16 through 18, determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. 16. Uxx + Uyy + Uzz = 0 17. Uxxxx +2uxxyy + Uyyyy = 0 18. ut + uux = 1 + Uxx 10. In each of Problems 19 through 21, verify that each given function is a solution of the given partial differential equation. 19. Uxx + Uyy = 0; u₁(x, y) = cos x cosh y, u₂(x, y) = ln(x² + y²) 20. a²uxx = ui; u₂(x, t) = e-α²x²t u₁(x, t) = e-a²t sinx, sin λx, À a real constant 21. a²uxx = Utti u₂(x, t) = sin(x-at), λ a real constant u₁(x, t) = sin(x) sin(at), 22. Follow the steps indicated here to derive the equation of motion of a pendulum, equation (12) in the text. Assume that the rod is rigid and weightless, that the mass is a point mass, and that there is no friction or drag anywhere in the system. to a. Assume that the mass is in an arbitrary displaced position, indicated by the angle 0. Draw a free-body diagram showing the forces acting on the mass. b. Apply Newton's law of motion in the direction tangential to the circular arc on which the mass moves. Then the tensile force in the rod does not enter the equation. Observe that you need to find the component of the gravitational force in the tangential direction. Observe also that the linear acceleration, as opposed to the angular acceleration, is Ld²0/dt², where L is the length of the rod. c. Simplify the result from part b to obtain equation (12) in the text.
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