21 ProblemsIn each of Problems 1 through 4, determine the order of the givendifferential equation; also state whether the equation is linear ornonlinear.1. 12d²y2.dy+1dt² dt3.+ 2y = sin t(1+ y²)dyd²ydt² dt+t-+y=e¹dªyd³y d²ydy+++ +y=1dt4 dt³ dt² dtd²y4.dt²In each of Problems 5 through 10, verify that each given function is asolution of the differential equation.5. y" -y=0; y(t) = e, y2(t) = cosht6. y" +2y' - 3y = 0; y₁(t) = e-³1, y₂(t) = et7. ty' - y = t²; y = 3t+t²8. y + 4y+3y= t; y₁(t) = t/3, y2(t) = e+t/39. 1²y" +5ty' +4y = 0, t 0; yi(t) = t2, y₂(t) = t-² Int10. y' - 2¹y = 1; y=e³²e³ds + e²+ sin(t + y) = sintIn each of Problems 11 through 13, determine the values of r for whichthe given differential equation has solutions of the form yet.11. y' +2y=012. y"+y'-6y=013. y" - 3y" + 2y' = 0In each of Problems 14 and 15, determine the values of r for which thegiven differential equation has solutions of the form y = t' for t 0.14. ty" +4ty' + 2y = 015. 1²y" - 4ty' + 4y = 0In each of Problems 16 through 18, determine the order of the givenpartial differential equation; also state whether the equation is linearor nonlinear. Partial derivatives are denoted by subscripts.16.Uxx + Uyy + Uzz = 017.Uxxxx +2uxxyy + Uyyyy = 018. ut + uux = 1 + Uxx10.In each of Problems 19 through 21, verify that each given function isa 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-α²x²tu₁(x, t) = e-a²t sinx,sin λx, À a real constant21. a²uxx = Uttiu₂(x, t) = sin(x-at), λ a real constantu₁(x, t) = sin(x) sin(at),22. Follow the steps indicated here to derive the equation of motionof a pendulum, equation (12) in the text. Assume that the rod is rigidand weightless, that the mass is a point mass, and that there is nofriction or drag anywhere in the system.toa. Assume that the mass is in an arbitrary displaced position,indicated by the angle 0. Draw a free-body diagram showing theforces acting on the mass.b. Apply Newton's law of motion in the direction tangential tothe circular arc on which the mass moves. Then the tensile forcein the rod does not enter the equation. Observe that you need tofind the component of the gravitational force in the tangentialdirection. Observe also that the linear acceleration, as opposed tothe angular acceleration, is Ld²0/dt², where L is the length ofthe rod.c. Simplify the result from part b to obtain equation (12) in thetext.

Question

21 ProblemsIn each of Problems 1 through 4, determine the order of the givendifferential equation; also state whether the equation is linear ornonlinear.1. 12d²y2.dy+1dt² dt3.+ 2y = sin t(1+ y²)dyd²ydt² dt+t-+y=e¹dªyd³y d²ydy+++ +y=1dt4 dt³ dt² dtd²y4.dt²In each of Problems 5 through 10, verify that each given function is asolution of the differential equation.5. y&#34; -y=0; y(t) = e, y2(t) = cosht6. y&#34; +2y' - 3y = 0; y₁(t) = e-³1, y₂(t) = et7. ty' - y = t²; y = 3t+t²8. y + 4y+3y= t; y₁(t) = t/3, y2(t) = e+t/39. 1²y&#34; +5ty' +4y = 0, t> 0; yi(t) = t2, y₂(t) = t-² Int10. y' - 2¹y = 1; y=e³²e³ds + e²+ sin(t + y) = sintIn each of Problems 11 through 13, determine the values of r for whichthe given differential equation has solutions of the form yet.11. y' +2y=012. y&#34;+y'-6y=013. y&#34; - 3y&#34; + 2y' = 0In each of Problems 14 and 15, determine the values of r for which thegiven differential equation has solutions of the form y = t' for t > 0.14. ty&#34; +4ty' + 2y = 015. 1²y&#34; - 4ty' + 4y = 0In each of Problems 16 through 18, determine the order of the givenpartial differential equation; also state whether the equation is linearor nonlinear. Partial derivatives are denoted by subscripts.16.Uxx + Uyy + Uzz = 017.Uxxxx +2uxxyy + Uyyyy = 018. ut + uux = 1 + Uxx10.In each of Problems 19 through 21, verify that each given function isa 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-α²x²tu₁(x, t) = e-a²t sinx,sin λx, À a real constant21. a²uxx = Uttiu₂(x, t) = sin(x-at), λ a real constantu₁(x, t) = sin(x) sin(at),22. Follow the steps indicated here to derive the equation of motionof a pendulum, equation (12) in the text. Assume that the rod is rigidand weightless, that the mass is a point mass, and that there is nofriction or drag anywhere in the system.toa. Assume that the mass is in an arbitrary displaced position,indicated by the angle 0. Draw a free-body diagram showing theforces acting on the mass.b. Apply Newton's law of motion in the direction tangential tothe circular arc on which the mass moves. Then the tensile forcein the rod does not enter the equation. Observe that you need tofind the component of the gravitational force in the tangentialdirection. Observe also that the linear acceleration, as opposed tothe angular acceleration, is Ld²0/dt², where L is the length ofthe rod.c. Simplify the result from part b to obtain equation (12) in thetext.

Accepted Answer

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21. a²uxx = ₁; u₁(x, t) Unti u₁(x, t) sin Ax, λ a real constant = sin(xx) sin(at), u₂(x, t) = sin(x-at), λ a real constant 22 F11