Differential Equations course

Model change with first-order, higher-order, and systems methods.

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7Core units

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Differential Equations course map: 7 units
#	Unit	What you can do after it	Weeks
01	Separable equations	You can split an equation so all the y parts sit on one side and all the x parts on the other, then integrate.	Weeks 1-2
02	Linear first-order equations	You can solve y' + p(x)y = q(x) by multiplying everything by an integrating factor.	Weeks 3-4
03	Second-order equations	You can solve equations containing a second derivative by finding the roots of a characteristic equation.	Weeks 5-6
04	Laplace transforms	You can turn a differential equation into ordinary algebra, solve it, and transform back.	Weeks 7-8
05	Systems	You can solve several linked differential equations at once using eigenvalues.	Weeks 9-10
06	Series solutions	You can build a solution as a power series when no neat formula exists, matching coefficients term by term.	Weeks 11-12
07	Qualitative analysis	You can describe how solutions behave without ever solving the equation, using slope fields and equilibrium points.	Weeks 13-14

Differential Equations course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Separable equations

Weeks 1-2

You can split an equation so all the y parts sit on one side and all the x parts on the other, then integrate. You can also use a starting value to pin down the constant.

Separate dy/dx = xy into dy/y = x dx.
Integrate both sides and then solve for y.
Apply a starting condition such as y(0) = 2.

Worked example

Solve dy/dx = xy with y(0) = 2

Separate: dy/y = x dx
Integrate: ln of the size of y = x squared / 2 + C

Answery = 2 times e to the power (x squared / 2)

Most common mistakeIntegrating dy/y as 1/y instead of a logarithm: that leads to y = 1/(x squared / 2 + C), which does not satisfy the original equation.

Explain separable equations Practice separable equations

Linear first-order equations

Weeks 3-4

You can solve y' + p(x)y = q(x) by multiplying everything by an integrating factor. You can also model a mixing tank or a simple circuit with one.

Compute the integrating factor as e to the integral of p.
Recognise the left side as a product derivative after multiplying.
Set up a salt-mixing tank equation from inflow and outflow rates.

Worked example

Solve y' + 2y = 6

Integrating factor: e to the 2x
The left side becomes the derivative of y times e to the 2x, so y e to the 2x = 3 e to the 2x + C

Answery = 3 + C e to the power -2x

Most common mistakeMultiplying only the left side by the integrating factor: that leaves y e to the 2x = 6x + C, which fails when substituted back in.

Explain linear first-order equations Practice linear first-order equations

Second-order equations

Weeks 5-6

You can solve equations containing a second derivative by finding the roots of a characteristic equation. You can also tell a spring that bounces from one that just settles.

Write and solve the characteristic equation for a constant-coefficient problem.
Use the sine and cosine form when the roots are complex.
Classify motion as underdamped, critically damped or overdamped from the roots.

Worked example

Solve y'' + 3y' + 2y = 0

Characteristic equation: r squared + 3r + 2 = 0
(r + 1)(r + 2) = 0, so r = -1 and r = -2

Answery = A e to the -x plus B e to the -2x

Most common mistakeKeeping only one root: writing y = A e to the -x leaves a single constant, which cannot be made to fit two initial conditions.

Explain second-order equations Practice second-order equations

Laplace transforms

Weeks 7-8

You can turn a differential equation into ordinary algebra, solve it, and transform back. You can also handle sudden switches and instant impulses.

Transform a second derivative using s squared Y minus s y(0) minus y'(0).
Use partial fractions before inverting a transform.
Handle a step input with the shifting rule.

Worked example

Transform y' + 3y = 0 with y(0) = 4

sY - 4 + 3Y = 0
Y(s + 3) = 4, so Y = 4/(s + 3)

Answery = 4 e to the power -3t

Most common mistakeDropping the initial value during the transform: writing sY + 3Y = 0 forces Y = 0, so the whole solution vanishes.

Explain laplace transforms Practice laplace transforms

Systems

Weeks 9-10

You can solve several linked differential equations at once using eigenvalues. You can also read a phase portrait to see where solutions end up.

Write a predator-prey pair as a single matrix system.
Use the eigenvalues to classify a node, a saddle or a spiral.
Sketch trajectories along the eigenvector directions.

Worked example

Classify the system x' = Ax where A has rows (1, 0) and (0, -2)

The eigenvalues are 1 and -2
One is positive and one is negative

AnswerA saddle point: solutions come in along one axis and shoot out along the other

Most common mistakeJudging stability from the matrix entries: the -2 suggests everything decays, but the eigenvalue 1 makes the origin unstable.

Explain systems Practice systems

Series solutions

Weeks 11-12

You can build a solution as a power series when no neat formula exists, matching coefficients term by term. You can also spot the points where the method breaks down.

Substitute a power series and shift indices so the powers line up.
Derive a recurrence relation for the coefficients.
Tell an ordinary point from a singular point.

Worked example

Solve y' = y as a power series with y(0) = 1

Write y = a0 + a1 x + a2 x squared + ... and match coefficients
a1 = a0, a2 = a1/2, a3 = a2/3, so a(n) = 1/n factorial

Answery = 1 + x + x squared/2 + x cubed/6 + ..., which is e to the x

Most common mistakeForgetting to shift the index after differentiating: terms in x to the n get matched against terms in x to the n-1, and the recurrence comes out wrong.

Explain series solutions Practice series solutions

Qualitative analysis

Weeks 13-14

You can describe how solutions behave without ever solving the equation, using slope fields and equilibrium points. You can also say which equilibria pull nearby solutions in.

Find equilibria by setting the derivative to zero.
Test stability from the sign of the derivative on each side.
Sketch a slope field and trace a solution curve through it.

Worked example

Find and classify the equilibria of dy/dt = y(1 - y)

Set y(1 - y) = 0, giving y = 0 and y = 1
Between 0 and 1 the derivative is positive, so solutions rise

Answery = 0 is unstable and y = 1 is stable

Most common mistakeCalling both equilibria stable because both are constant solutions: a solution starting at y = 0.1 walks away from 0 and climbs toward 1.

Explain qualitative analysis Practice qualitative analysis

Prepare for Differential Equations practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Separable equations from a blank page. Then connect it to Linear first-order equations. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Calculus calculator can test calculations and representations used in Differential Equations. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Differential Equations practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesCalculus explainedConcepts, worked methods and checks behind this course. PracticeMixed Differential Equations questionsAnswer review after every attempt. ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Questions about the Differential Equations course

Where should I start?

Start with Separable equations if you are following the full sequence. If that unit feels automatic, open the Differential Equations practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Differential Equations test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Calculus calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Differential Equations course

Units at a glance

Differential Equations course units

Separable equations

Worked example

Linear first-order equations

Worked example

Second-order equations

Worked example

Laplace transforms

Worked example

Systems

Worked example

Series solutions

Worked example

Qualitative analysis

Worked example

Prepare for Differential Equations practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other university courses

Discrete Mathematics

Probability

Mathematical Statistics

Take the complete Differential Equations test

Review essential formulas

Use the Calculus calculator

Prepare around your exam date

Questions about the Differential Equations course