The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1023: Multivariable Calculus and Modelling Semester 2, 2023
This individual assignment is due by 11:59pm Wednesday, 23 August, 2023,
via Canvas. Late assignments will receive a penalty of 5% per day until the closing
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but students must write up and submit their own version of the solutions.
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Copyright c© 2023 The University of Sydney 1
The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH1023: Multivariable Calculus and Modelling Semester 2, 2023
1. Calculate the solution of the differential equation
y2
dy
dx
− x2(1 + y3) = 0,
satisfying the condition y(0) = 2.
2. Newton’s Law of Cooling states that the rate of change of the temperature T of an object
is proportional to the difference between T and the temperature τ of the surrounding
medium:
dT
dt
= −k(T − τ).
where k is a positive constant.
(a) Solve this equation for T given that T (0) = T0.
(b) Take T0 > τ . What is the limiting temperature to which the object cools as t
increases? What happens if T0 < τ?
3. Water is pumped into a tank to dilute a saline solution. The volume of the solution, V ,
is kept constant by continuous outflow. The amount of salt in the tank, s, depends on
the amount of water that has been pumped in, denoted by x. Given that
ds
dx
= − s
V
,
find the amount of water that must be pumped into the tank to eliminate 50% of the
salt. Take V as 10,000 litres and assume s(0) = s0.
Questions 4 & 5 – Next page
2
4. Find the general solution y of the first-order equation
2xyy′ = y2 − x2.
Hint:
Use the substitution y = ux to transform the equation into a first-order, separable,
differential equation for u.
5. Find the general solution of the first-order differential equation
e−x(−e−y + 1)dy + e−ydx = 0.