Engineering Mathematics Solved Problems

(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.

(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.(1) Which of the following sets are linearly independent and which are linearly dependent?

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(c) Calculate the inverse Laplace transform of I 2 (s).

[8] (d)Calculatei 1 (t) by insertingi 2 (t) and its derivative into the original equation.

If you are having difficulties with these questions practice using the remaining questions from these exercise.

You will get the best from this tutorial by working though these examplesbeforethe tutorial and asking your tutors to help you with questions with which you are having problems.

[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.

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.