What does this Differential Equations cheat sheet cover?

It has 10 sections: 1st-order basics, Separable / exact, Existence, 2nd-order form, Root cases, Nonhomogeneous, Variation, Laplace, Systems and Method picks. Together they cover the topics a typical Differential Equations exam tests, including separable equations, first-order linear (integrating factor), exact equations and second-order linear homogeneous with constant coefficients.

Is the differential equations cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Differential Equations cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Differential Equations sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathDifferential Equations

Differential Equations Cheat Sheet — 1-page PDF

Separable, linear, exact, 2nd-order, Laplace, systems, existence/uniqueness. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Differential Equations Final exam study sheet.

1 pageUS Letter
10 sections10 topics
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Preview of the one-page Differential Equations cheat sheet: Separable, linear, exact, 2nd-order, Laplace, systems, existence/uniqueness — the whole course, one page

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What’s on it

The topics a typical Differential Equations exam tests, all on the one page.

Separable equations
First-order linear (integrating factor)
Exact equations
Second-order linear homogeneous with constant coefficients
Characteristic equation cases
Undetermined coefficients
Variation of parameters
Laplace transform table and properties
Systems of ODEs with eigenvalues
Existence and uniqueness

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The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

011st-order basics

ODE: Equation with unknown function $y (x)$ and derivatives in one independent variable $x$ .
Order / linear: Order = highest derivative. Linear iff $y, y^{'}, \dots$ appear to first power, no products, coeffs depend only on $x$ .

Separable

\frac{d y}{d x} = g (x) h (y) \Rightarrow \int \frac{1}{h ( y )} d y = \int g (x) d x + C

Valid when $h (y) \neq = 0$ on interval; constant solutions from $h (y) = 0$ may be missed.

Separate $y$ -terms and $x$ -terms.
Integrate both sides.
Apply initial condition if given.
Check constant / lost solutions.

1st-order linear

y^{'} + p (x) y = q (x)

Standard form.

μ (x) = e^{\int p (x) d x}

Integrating factor for $y^{'} + p (x) y = q (x)$ .

Linear solution

(μ y)^{'} = μ q \Rightarrow y = \frac{1}{μ} (\int μ q d x + C)

Multiply ODE by $μ$ .

02Separable / exact

Exact equation: $M (x, y) d x + N (x, y) d y = 0$ is exact iff $\partial M / \partial y = \partial N / \partial x$ on a region.

03Existence

Condition	Guarantee	Typical check
$f$ continuous	Existence	No blow-up / discontinuity
$f_{y}$ continuous	Uniqueness	No cusp / vertical slope dependence
Either fails	May fail	Need separate analysis

Watch out:Continuity of $f$ does not imply uniqueness. Nonunique example often starts from $y^{'} = ∣ y ∣$ at $y (0) = 0$ .

Interval

Local solution exists on some open interval containing x $_{0};$ maximal interval can end where f or f $_{y}$ blows up or is undefined.

Watch domain restrictions.

042nd-order form

Homogeneous 2nd-order: $a y^{''} + b y^{'} + cy = 0$ with constants $a \neq = 0, b, c$ .

Characteristic

a r^{2} + b r + c = 0

Assume trial $y = e^{r x}$ .

Distinct real roots

r_{1} \neq = r_{2} \in R \Rightarrow y = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}

Homogeneous constant-coefficient case.

Repeated root

r_{1} = r_{2} = r \Rightarrow y = (c_{1} + c_{2} x) e^{r x}

Multiplicity $2$ .

Complex roots

r = α \pm i β \Rightarrow y = e^{α x} (c_{1} cos β x + c_{2} sin β x)

$β > 0$ .

05Nonhomogeneous

Undetermined coeffs

$y_{p} =$ trial of same family as $g (x)$

Use only when $g$ is polynomial, $e^{a x}$ , $sin b x$ , $cos b x$ , or finite sums/products of these.

$g (x)$	Trial $y_{p}$
Polynomial degree $n$	$A_{n} x^{n} + \dots + A_{0}$
$e^{a x}$	$A e^{a x}$
$sin b x, cos b x$	$A cos b x + B sin b x$
Product	Multiply corresponding trials

Resonance

If trial term overlaps $y_{h},$ multiply trial by $x^{s}$

$s$ = smallest integer making independence; typically multiplicity of root in characteristic equation.

General solution

y = y_{h} + y_{p}

For linear ODEs.

Watch out:Method fails for non-polynomial/non-exponential forcing like $tan x$ , $ln x$ unless rewritten or another method used.

Ex.Find a trial for $y^{''} - y = e^{x}$ .

$y_{h}$ has root $r = \pm 1$ , so $e^{x}$ resonates. Trial $y_{p} = A x e^{x}$ .

06Laplace

Convolution

L^{- 1} {F (s) G (s)} = (f * g) (t) = \int_{0}^{t} f (τ) g (t - τ) d τ

Useful when product in $s$ -domain.

Ex.Solve $y^{'} + y = 1$ , $y (0) = 0$ by Laplace.

$s Y - y (0) + Y = 1/ s$ ⇒ $(s + 1) Y = 1/ s$ ⇒ $Y = 1/ [s (s + 1)]$ . Invert: $Y = 1/ s - 1/ (s + 1)$ ⇒ $y = 1 - e^{- t}$ .

07Systems

Linear system: $x^{'} = A x$ with constant matrix $A$ ; scalar ODEs coupled in vector form.

Eigen-solution

x (t) = e^{λ t} v if A v = λ v

For eigenpair $(λ, v)$ .

General real solution

x (t) = \sum c_{k} e^{λ_{k} t} v_{k}

For independent eigenvectors; use real/imag parts for complex pairs.

Eigenvalues	Solution form	Notes
Distinct real $λ_{1}, λ_{2}$	$c_{1} e^{λ_{1} t} v_{1} + c_{2} e^{λ_{2} t} v_{2}$	Diagonalizable
Complex $α \pm i β$	$e^{α t} (c_{1} cos β t + c_{2} sin β t)$	Real-valued basis
Repeated $λ$	Need generalized eigenvector if not enough eigenvectors	Jordan form

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Also on the full Differential Equations sheet

9 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Separable / exact

Potential
Separable IC

Existence

IVP
Existence
Uniqueness

2nd-order form

Root cases

Euler trial
Wronskian test

Nonhomogeneous

Forced ODE

Variation

Variation of parameters
Auxiliary conditions
Solve for ui'
Integrate

Laplace

Transform
Key properties
Shift
Step shift

Systems

Method picks

IC handling

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FAQ

Questions

About the Differential Equations sheet and making your own.

What does this Differential Equations cheat sheet cover?: It has 10 sections: 1st-order basics, Separable / exact, Existence, 2nd-order form, Root cases, Nonhomogeneous, Variation, Laplace, Systems and Method picks. Together they cover the topics a typical Differential Equations exam tests, including separable equations, first-order linear (integrating factor), exact equations and second-order linear homogeneous with constant coefficients.
Is the differential equations cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Differential Equations cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Differential Equations sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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