Differential equations

Solve differential equations of the form

\frac{dy}{dx} = f(x)

by direct integration and separable equations of the form

\frac{dy}{dx} = g(x)h(y)

; interpret direction fields qualitatively; model and solve simple problems with first-order differential equations.

Worked examples

Direct integration (simplest type)

Straightforward

Problem

Solve

\dfrac{dy}{dx} = 4x^3 - 2

with

y(1) = 3

Separable equation

Moderate

Problem

Solve

\dfrac{dy}{dx} = \dfrac{y}{x}

with

y(1) = 5

Mixture / flow problem

Challenging

Problem

A tank contains 50 L of pure water. Brine containing 3 g/L flows in at 2 L/min; the well-mixed solution drains at 2 L/min. Set up and solve the differential equation for the salt content

Q(t)

(grams) at time

t

minutes.

Practise

Open Math

Differential equations

Calculus · ME-12-05

Name:

Date:

Q1Straightforward

Which of the following is the general solution of

\dfrac{dy}{dx} = 3x^2

A. $y = x^3 + C$
B. $y = 6x + C$
C. $y = x^3$
D. $y = \dfrac{x^3}{3} + C$

Q2Straightforward

The differential equation

\dfrac{dy}{dx} = 2x + 1

has general solution

y = x^2 + x + C

. If

y = 5

when

x = 1

, find the value of

C

Q3Straightforward

A differential equation is called **separable** if it can be written in the form

\dfrac{dy}{dx} = g(x)h(y)

. Which of the following is separable?

A. $\dfrac{dy}{dx} = \dfrac{x}{y}$
B. $\dfrac{dy}{dx} = x + y$
C. $\dfrac{dy}{dx} = \sin(x+y)$
D. $\dfrac{dy}{dx} = x^2 + y^2$

Q4Straightforward

The differential equation

\dfrac{dy}{dx} = e^x

with initial condition

y(0) = 2

has solution

y = e^x + C

. Find the value of

C

Q5Moderate

Solve the separable equation

\dfrac{dy}{dx} = 2xy

with initial condition

y(0) = 3

. Find

y(1)

to 4 decimal places.

Q6Moderate

Solve

\dfrac{dy}{dx} = \dfrac{x}{y}

with

y(0) = 2

. Find

y(3)

to 4 decimal places.

Q7Moderate

A direction field (slope field) for

\dfrac{dy}{dx} = y

shows line segments with positive slope wherever

y > 0

and negative slope wherever

y < 0

. Which function family is the general solution?

A. $y = Ce^x$
B. $y = Cx$
C. $y = C\ln x$
D. $y = x^2 + C$

Q8Moderate

The rate of change of a quantity

P

satisfies

\dfrac{dP}{dt} = 5 - P

. Find the general solution

P(t)

. If

P(0) = 2

, what is

P(\ln 2)

to 2 decimal places?

Q9Moderate

Solve

\dfrac{dy}{dx} = \cos x

with

y(0) = 1

. Find

y\!\left(\dfrac{\pi}{2}\right)

Q10Challenging

Solve the separable equation

\dfrac{dy}{dx} = \dfrac{1-y^2}{y}

with

y(0) = 2

. Find

y(1)

to 4 decimal places.

Q11Challenging

A tank initially contains 100 L of pure water. Salt solution containing 2 g/L flows in at 5 L/min, and the well-mixed solution drains at 5 L/min. The amount of salt

Q

grams after

t

minutes satisfies

\dfrac{dQ}{dt} = 10 - \dfrac{Q}{20}

. If

Q(0) = 0

, find

Q(20)

to 2 decimal places.

Q12Challenging

The velocity

v

(m/s) of a particle satisfies

\dfrac{dv}{dt} = 4 - v

with

v(0) = 0

. Find the time

t

(in seconds, to 4 decimal places) at which

v = 2

Worked solutions and answers at openmath.au/year-12/extension-1/further-applications-of-calculus/differential-equations