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[20] SCEE08009 Engineering Mathematics 2A Tutorial 2 (d) ( d 3 x dt 3 t d 2 x dt 2 x 2 t= sint, x(1) = 1,x ̇(1) = 0,x ̈(1) =− 2. 2 ,x ̇(1) = 1,x ̈(1) = 0, using forward Euler with a step size of ∆t= 0.025s.

Repeat the calculation with ∆t= 0.0125 and hence estimate the accuracy of your solution att= 2.

(a) dx dt −kx= 0 (b) d 3 x dt 3 t 2 dx dt =t d dt (xt) (d) d dt t d dt (t 2 x) =xt.

(3) Determine which members of the given sets are solutions of the differential equation.

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The topics are Chain rule, Partial Derivative, Taylor Polynomials, Critical points of functions, Lagrange multipliers, Vector Calculus, Line Integral, Double Integrals, Laplace Transform, Fourier series.Page 2 SCEE08009 Engineering Mathematics 2A Tutorial 2 h 0 2 sinωt.whereh 0 is the height of the waves acting on the float andωis the frequency of the waves.Hence write down the general solution of the differential equation (a) d 2 x dt 2 −p 2 x= 0 dx dt d 2 x dt 2 dx dt x= 0 dx dt (1) The temperature, T, (in Kelvin) of a reaction vessel during a particularly unstable chemical reaction is modelled by the linear differential equation, d 4 T dt 4 d 3 T dt 3 d 2 T dt 2 d T dt At the start of the experiment the reactor vessel is at 293K.Assuming, at the start of the reaction, the first derivative −20 Ks− 1 , the second derivative is−10 Ks− 2 and all other derivatives are zero, determine the temperature of the reaction vessel after 3 seconds.The analytical solution to this BVP is x=e 3 t 2 (28π 3 35π) sint 320 π 4 800π 2 3380 21 πcost 80 π 4 200π 2 845 (28π 2 −91) sinπt− 84 πcosπt 320 π 4 800π 2 3380 Calculate the predicted position,x 50 , which is found by performing 50 time steps with the forward Euler ODE solver with a uniform time step, ∆t= 0.1s.By comparing x 50 with the analytical solution,x(5), comment on the accuracy of your solution.In a particular experiment the float is excited by 700mm, 2s period, waves.The float is at rest at the start of the experiment sox(0) = 0.0 and ̇x(0) = 0.0.F 1 (s) = s 3 , ℜ(s) Remark:αandβare positive and real constants.(a)Calculate the Laplace transform - Use the Laplace transform properties [10] from Table 1 and the function transforms from Table 3 to calculate the Laplace transform of the following time domain function.

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