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MAST10006 Calculus
Assignment
• Answer all the questions below. Of these questions, one will be chosen for marking.
• Submit your assignment in Canvas as a single PDF file before the deadline above.
See Canvas for instructions on how to submit the assignment.
• Marks may be awarded for:
◦ Correct use of appropriate mathematical techniques.
◦ Accuracy and validity of any calculations or algebraic manipulations.
◦ Clear justification or explanation of techniques and rules used.
◦ Clear communication of mathematical ideas through diagrams
◦ Use of correct mathematical notation and terminology.
• You must explicitly state the techniques used in your answers when evaluating limits.
• You must use methods taught in MAST10006 Calculus 2 to solve the assignment questions.
• Give your answers as exact values.
1. (a) Solve the initial valued problem (in R)
3y′′ + 6y′ + 15y = 0, y(0) = 2, y′(0) = 2. (1)
(b) Rewrite the solution of (1) under the form
y(x) = Cef(x) cos(g(x))
where C is a real number and f(x), g(x) are polynomials of x to be determined by
you.
(c) Sketch the graph of the solution of (1).
2. (a) Find the general solutions (in R) of the homogeneous ODE:
y′′ + 2y′ − 3y = 0 (2)
(b) Using your answer to question (2a), find the general solutions (in R) of the inhomo-
geneous ODEs:
i. y′′ + 2y′ − 3y = 5 cosh(4x)
ii. y′′ + 2y′ − 3y = 6ex
iii. y′′ + 2y′ − 3y = 3x2 − ex
End of assignment