The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH1023: Multivariable Calculus and Modelling Semester 2, 2023
This individual assignment is due by 11:59pm Wednesday 11 October 2023, via
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This assignment is worth 10% of your final assessment for this course. Your answers should be
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Copyright c© 2023 The University of Sydney 1
The University of Sydney
School of Mathematics and Statistics
Assignment 2
MATH1023: Multivariable Calculus and Modelling Semester 2, 2023
1. Find the General Solution and the Particular Solution to the following differential equa-
tion:
dy
dx
− (sinhx)y = (3x2)ecoshx, y(0) = e
(All steps in the calculations must be clearly shown.)
2. Consider the second-order differential equation
d2x
dt2
+ a
dx
dt
+ b2x = 0 (a, b positive real constants)
(a) Find the characteristic equation and determine the conditions on a, b to obtain
subcritical damping.
(b) If a = b = 2, find the particular solution which satisfies x = 3 and
dx
dt
= −2 when
t = 0.
(All steps in the calculations must be clearly shown.)
3. Consider the surface defined by z = f(x, y), where:
f(x, y) =
x2y
2(x2 + y2)
Write down the equation of the tangent planes to the surface f(x, y) at the points
(a) x = 1, y = −1
(b) x = 2, y = 2
(All steps in the calculations must be clearly shown.)
4. Consider the surface defined by z = f(x, y), where
f(x, y) = (x + 1)2 + (y − 1)2 + 4
(a) Draw the level curves of the function at c = 4, c = 8, c = 13.
(b) Draw a rough sketch of the function. (No need to draw the level curves here)
5. Consider the parametric equation
x = 2 cos t, y =
sin t
9
, z = t2.
(a) Describe the curve in 3 dimensions and draw a rough sketch of it.
(b) Describe the behaviour of the curve as t increases.