Calculus I Key Points Summary

A complete summary of Calculus I topics for quick review and memorization

Limits and continuity

Limits of sequences

Sequences

Definition of a sequence

A function on the set of positive integers $N$: $f=x_{n}$
General term: $x_{n}$

Monotonicity of sequences

Boundedness of sequences

Bounded sequence: for sequence {${x_{n}}$}, there exists $M>0$ and always $|x_{n}|\leq M$. It has both an upper bound and a lower bound.

Sequence limits

• Convergence: if when $n$ grows without bound, the general term $x_{n}$ of the sequence {${x_n}$} can approach a fixed constant $A$, then $A$ is the limit and {${x_{n}}$} converges; otherwise it diverges.
  Example: -1, 1, -1, 1… is divergent
  Example: $x_{n}=\frac{n-1}{n}$ the -1 can be ignored because it is tiny, so the limit is 1

Properties of convergent sequences

• Uniqueness: if sequence {$x_{n}$} converges, then its limit is unique (not vice versa)

  • -> Unbounded implies divergence

• Boundedness: every convergent sequence is bounded

• Sign preservation: in a neighborhood between 0 and the limit, if it is positive at one point, then it stays positive (and vice versa)

Limits of single-variable functions

Concept of function limits

Limits as the variable approaches infinity

• $\lim_{ x \to \infty }f(x)=A$
  If it exists, it converges; otherwise it diverges.
  Consider positive and negative infinity separately.

• $\lim_{ x \to +\infty }f(x)=A$ and $\lim_{ x \to -\infty }f(x)=A \iff \lim_{ x \to \infty }f(x)=A$ (two-sided limit)

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f(x)=1/x

Limits as the variable approaches a finite value

Not necessarily equal to the function value.
Example:
$$
f(x) = \begin{cases}
x+1, & x < 1 \
3 & x = 1 \
3x-1 & x>1
\end{cases}
$$
When $x\to 1$, the limit is 2, but the function value is 3 (because it never actually reaches 1)

• Left-hand and right-hand limits ($x\to x^+$ and $x\to x^-$)

• $\lim_{ x \to +x_{0}^- }f(x)=A$ and $\lim_{ x \to x_{0}^+ }f(x)=A \iff \lim_{ x \to x_{0} }f(x)=A$
  i.e. left and right must be equal for the limit to exist

Properties of function limits

• Uniqueness: if the limit exists, it is unique

• Local boundedness

• Local sign preservation

Limit operations for single-variable functions

Limits of arithmetic operations

$$
\begin{aligned}
&\lim [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x) \
&\lim [f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x) \
&\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}, \quad \lim g(x) \neq 0
\end{aligned}
$$

This is the standard statement of the limit rules for arithmetic operations.

• Corollaries:

• $\lim[f(x)]^n=lim[f(x)]^n$

• $\lim[Cf(x)]=CA$

• When computing, make sure all function limits exist and are meaningful

Limits of composite functions

If $\lim_{ x \to x_{0} }g(x)=u_{0}$ and $\lim_{ u \to u_{0} }f(u)=A$, then $\lim_{ x \to x_{0}} f[g(x)]=\lim_{ u \to u_{0 }} f(u)=a$
Compute limits from inside to outside, and the inner limit value becomes the outer input.

Two important limits

• Limit existence criterion 1: during $x\to x_{0}$, the functions $f(x)$, $g(x)$, and $h(x)$ are defined, and satisfy $g(x)\le f(x)\leq h(x)$ and $\lim_{ x \to x_{0} } = \lim_{ x \to x_{0} }=a$, then $\lim_{ x \to x_{0} }=A$ (A on both sides implies A in the middle)

• Important limit 1 (squeeze theorem) # memorize
  $$\lim_{ x \to 0 } \frac{\sin[x]}{x}=1$$

• Limit existence criterion 2: a monotone bounded sequence must have a limit

• Important limit 2: # memorize
  $$\lim_{ x \to \infty } \left( 1+\frac{1}{x} \right)^x=e$$

• Supplement

• $$\lim_{ x \to x_{0} } \frac{a_{0}x^m+a_{1}x^{m-1}……}{b_{0}x^n+b_{1}x^{n-1}……}=\begin{cases} \frac{a_{0}}{b_{0} }& n=m \ 0 & n>m \ \infty & n<m\end{cases}$$

Infinitesimals and infinities

Concepts

• Infinitesimal: if as $x\to x_{0}$ the limit of $f(x)$ is 0, then $f(x)$ is infinitesimal at $x\to x_{0}$ (not negative infinity, but 0)

• Infinity: … is $\infty$, divided into positive infinity and negative infinity

• If $f(x)$ is infinite, $\frac{1}{f(x)}$ is infinitesimal. If $f(x)$ is infinitesimal and $f(x)\neq 0$, then $\frac{1}{f(x)}$ is infinite.

Properties of infinitesimals

• The sum of finitely many infinitesimals is infinitesimal

• The product of a bounded function and an infinitesimal is infinitesimal

• The product of a constant and an infinitesimal is infinitesimal

• The product of finitely many infinitesimals is infinitesimal

Comparing infinitesimals

Concept of comparison

$\beta$ and $\alpha$ are infinitesimals

• Higher-order infinitesimal ($\beta=o$): $\lim \frac{\beta}{\alpha}=0$ ($\beta$ is a higher-order infinitesimal than $\alpha$)

• Lower-order infinitesimal: $\lim \frac{\beta}{\alpha}=\infty$

• Same order infinitesimal: $\lim \frac{\beta}{\alpha}=C (C\neq 0)$

  • Equivalent infinitesimal ($\alpha \sim \beta$): $\lim \frac{\beta}{\alpha}=1$

    Application

    Equivalent infinitesimal relations: # memorize
    $$\begin{align} \sin x \sim x \ \arcsin x \sim x \ \tan x \sim x\ \arctan x \sim x \ e^x -1\sim x \ a^x-1 \sim x\ln a(a>0) \ 1-\cos x \sim \frac{1}{2}x^2 \ \ln(1+x) \sim x \ (1+x)^a-1 \sim ax(a\neq 0) \end{align}$$
    When $a$, $a’$, $b$, and $b’$ are infinitesimals, $aa’$, $bb’$, and $lim \frac{a’}{b’}$ exists or is infinite
    $$\lim \frac{a}{b} = \lim \frac{a’}{b’} $$

Continuity of single-variable functions

Concept of continuity

Definition

Defined + limit exists as $x\to x_{0}$ + $\lim_{ x \to x_{0} }f(x)=f(x_{0})$ = the function is continuous at $x_{0}$
Left continuity, right continuity

Continuity on an interval

Discontinuities and types

Discontinuous, # memorize

$$
Discontinuities\begin{cases}
First kind \begin{cases} Jump discontinuity: left and right limits exist but are not equal \ Removable discontinuity: not defined at $x_{0}$ or $\lim_{ x \to x_{0} }f(x)\neq f(x_{0}){}\end{cases}\
Second kind \begin{cases} Infinite discontinuity: at least one limit is infinite \ Oscillatory discontinuity: at least one limit does not exist\end{cases}
\end{cases}
$$

Operations and properties of continuous functions

Operations and continuity of elementary functions

Rules for arithmetic operations

If two single-variable functions are continuous at $x_{0}$, then the sum, difference, product, and quotient are still continuous.

Continuity of inverse functions

If a single-variable function is monotonic and continuous, then its inverse is also continuous.

Continuity of composite functions

$\lim_{ x \to x_{0}}(x)=u_{0}$, $y=f(u)$ is continuous at $u_{0}$, and $y=f[g(x)]$ is defined. Then as $x\to x_{0}$, $y=f[g(x)]$ has a limit and $\lim_{ x \to x_{0} }f[g(x)]=\lim_{ u \to u_{0} }f[u]=f(u_{0})$
That is, $\lim_{ x \to x_{0}}f[g(x)]=f[\lim_{ x \to x_{0} }g(x)]$
All elementary functions are continuous on their domains

Properties of continuous functions on a closed interval

• Boundedness and extreme value theorem: a single-variable continuous function defined on a closed interval is bounded on that interval and must attain a maximum and a minimum (not vice versa)

• Intermediate value theorem: if one value is positive and one is negative, there must be a zero in between

• Value range theorem: a generalization of the intermediate value theorem

Derivatives and differentials

Derivative concepts

Definition

The derivative is written as $f’(x)=y’=\frac{dy}{dx}=\frac{df(x)}{dx}$

Examples # memorize

Common derivative formulas

• $(C)’ = 0$;

• $(x^\mu)’ = \mu x^{\mu - 1}$;

• $(\sin x)’ = \cos x$;

• $(\cos x)’ = -\sin x$;

• $(\tan x)’ = \sec^2 x$;

• $(\cot x)’ = -\csc^2 x$;

• $(\sec x)’ = \sec x \tan x$;

• $(\csc x)’ = -\csc x \cot x$;

• $(a^x)’ = a^x \ln a \ (a > 0, a \neq 1)$;

• $(e^x)’ = e^x$;

• $(\log_a x)’ = \frac{1}{x \ln a} \ (a > 0, a \neq 1)$;

• $(\ln x)’ = \frac{1}{x}$;

• $(\arcsin x)’ = \frac{1}{\sqrt{1 - x^2}}$;

• $(\arccos x)’ = -\frac{1}{\sqrt{1 - x^2}}$;

• $(\arctan x)’ = \frac{1}{1 + x^2}$;

• $(\text{arccot } x)’ = -\frac{1}{1 + x^2}$.

One-sided derivatives

Left derivative, right derivative
Differentiable $\leftrightarrow$ left and right derivatives exist and are equal

Geometric meaning of the derivative # memorize

Tangent line equation: $y-y_{0}=f’(x_{0})(x-x_{0})$
Normal line (perpendicular to the tangent) equation: $y-y_{0}=-\frac{1}{f’(x_{0})}(x-x_{0})$

Relationship between differentiability and continuity

Differentiable $\to$ continuous (not vice versa)

Rules of differentiation

Basic differentiation rules # memorize

$$\begin{align} (u\pm v)’=u’\pm v’ \ (Cu)’=Cu’ \ (u_{1}u_{2}\dots)’=u_{1}’u_{2}\dots u_{n}+u_{1}u_{2}’\dots u_{n}+\dots \ \left( \frac{u}{v} \right)’=\frac{u’v-uv’}{v^2}\end{align}$$

Differentiation of composite functions

Chain rule: differentiate from outside to inside, $dy/dx$ is the derivative of $y$ with respect to $x$
$$\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx} = f’(u)\cdot g’(x)$$

Differentiation of inverse functions

If $x=\phi(y)$ is monotonic, continuous, and differentiable, then the inverse function $y=f(x)$ is differentiable, and
$$f’(x)=\frac{1}{\phi’(y)} / \frac{dy}{dx} = \frac{1}{\frac{dy}{dx}}$$
Derivative of inverse = reciprocal of derivative of original

Higher-order derivatives

Just differentiating multiple times.
The $n$th derivative is written as $y^{(n)}、f^{(n)}、 \frac{d^ny}{dx^n}$
Higher-order derivative rules: # memorize
$$(C_{1}u\pm C_{2}v)^{(n)}=C_{1}u^{(n)}\pm C_{2}v^{(n)}$$
$$(uv)^{(n)}=\sum_{i=1}^nC^k_{n} u^{(k)}v^{(n-k)}$$ (Leibniz formula)

Derivatives of implicit and parametric functions

Implicit derivatives

A function that cannot be expressed directly as $y=…$
Differentiate both sides, and use the chain rule. Keep $y’$.
Logarithmic differentiation: take the logarithm of both sides, often used for exponential functions: $x^{\sin x}=\sin x\ln x$

Derivatives of parametric equations

$$
\begin{cases}
x = x(t)\
y = y(t)
\end{cases}
$$

Take the inverse of the first and substitute into the second
$$\frac{dy}{dx}=\frac{\frac{dy}{dt}} {\frac{dx}{dt}}$$

Differentials

Concept of differential

Assume a function is defined on an interval. For a point $x_0$ and any increment $\Delta x$ such that $x_0+\Delta x$ is still within the interval, the increment of the function can be written as

$$
\Delta y = A,\Delta x + o(\Delta x),
$$

where $A$ is a constant and $o(\Delta x)$ is a higher-order infinitesimal compared to $\Delta x$. Then $y=f(x)$ is differentiable at $x_0$, and $A\Delta x$ is the differential of $y=f(x)$ at $x_0$ corresponding to the increment $\Delta x$, written as $dy = A,\Delta x$. It is the linear principal part.
Single-variable function differentiable $\leftrightarrow$ differential exists

Invariance of differentials

Whether $u$ is the independent variable or an intermediate variable, the differential $dy$ of $y=f(u)$ can always be written as $f’(x)$ times $du$.

Basic differential formula

$\frac{dy}{dx}=f’(x)$, so
add $dx$ at the end
Only one $dx$ may remain in the differential result.

Mean value theorems and applications of derivatives

Mean value theorems

Rolle’s theorem

Fermat’s lemma

Function $f(x)$ is defined in some neighborhood $U(x_{0})$ of $x_{0}$ and is differentiable at $x_{0}$. If for any $x\in U(x_{0})$ we have $f(x)\leq f(x_{0}) / f(x)\geq f(x_{0})$, then $f’(x)=0$.

Rolle’s theorem

If $f(x)$ satisfies
(1) continuous on the closed interval $[a,b]$ (closed-continuous, open-differentiable)
(2) differentiable on the open interval $(a,b)$
(3) equal values at endpoints ($f(a)=f(b)$) (same height at both ends)
Then there exists at least one point $t\in(a,b)$ such that $f’(t)=0$.

Lagrange mean value theorem

• Finite increment theorem
  If $f(x)$ satisfies
  (1) continuous on the closed interval $[a,b]$
  (2) differentiable on the open interval $(a,b)$
  Then there exists at least one point $t\in(a,b)$ such that $f(b)-f(a)=f’(t)(b-a)$

    <span style="color:rgb(255, 92, 92)">the slope at some point equals</span> the slope of the $ab$ <span style="color:rgb(255, 92, 92)">line segment</span>
  

• Finite increment formula
  $\Delta y=f’(x+\theta \Delta x)\cdot \Delta x$

• Corollary 1: if the derivative of $f(x)$ on interval $I$ is always zero, then $f(x)$ is a constant on $I$

• Corollary 2: if $f(x)$ and $g(x)$ satisfy $f’(x)=g’(x)$ on interval $I$, then $f(x)=g(x)+C$ on $I$

L’Hopital’s rule

Indeterminate forms 0/0 and $\infty/\infty$

memorize

• Theorem 3:
  (1) as $x\to a$, functions $f(x)$ and $g(x)$ both approach 0
  (2) in a punctured neighborhood of $a$, $f’(x)$ and $g’(x)$ exist and $g’(x)\neq 0$
  (3) $\lim_{ x \to a } \frac{f’(x)}{g’(x)}$ exists or is infinite
  Then $\lim_{ x \to a } \frac{f(x)}{g(x)} = \lim_{ x \to a } \frac{f’(x)}{g’(x)}$
  You can apply the theorem repeatedly until it is no longer indeterminate.

• Theorem 4 (corollary)
  (1) as $x\to \infty$, functions $f(x)$ and $g(x)$ both approach 0
  (2) for sufficiently large $|x|$, $f’(x)$ and $g’(x)$ exist and $g’(x)\neq 0$
  (3) $\lim_{ x \to a } \frac{f’(x)}{g’(x)}$ exists or is infinite
  Then $\lim_{ x \to \infty } \frac{f(x)}{g(x)} = \lim_{ x \to \infty } \frac{f’(x)}{g’(x)}$

Other indeterminate forms

Convert to a standard form.

Monotonicity and concavity

Monotonicity test

Derivative $>0 \leftrightarrow$ increasing, and vice versa

Concavity test

Second derivative $>0 \leftrightarrow$ concave, and vice versa (slope increasing means concave)

Extrema and extreme values

Extrema and how to find them

• The concept of an extremum is local; derivative equals 0, but not vice versa

• Suppose $f$ is continuous at $x_{0}$ and differentiable on a punctured neighborhood $U(x_{0},t)$

  • When $x\in(x_{0}-t,x_{0})$, $f’(x)>0$, and when $x\in(x_{0},x_{0}+t)$, $f’(x)<0$, $f(x)$ has a maximum at $x_{0}$ (and vice versa, left increasing right decreasing)

• If $f’(x_{0})=0$

  • When $f’’(x_{0})<0$, $f(x)$ has a local maximum at $x_{0}$ (concave and extreme)

Optimization problems

Indefinite integrals

Concepts and properties of indefinite integrals

Concept of an antiderivative

An antiderivative is a function whose derivative is the given function.

• An antiderivative is not unique; the difference is a constant

• A continuous function always has an antiderivative

Concept of indefinite integral

Let $F(x)$ be an antiderivative of $f(x)$ on interval $I$. Then all antiderivatives of $f(x)$ on $I$, $F(x)+C$, are called the indefinite integral of $f(x)$, written as $\int f(x)dx$, with $x$ the integration variable.
Integration and differentiation are inverse operations

Basic integral formulas # memorize

• $\int k\mathrm{d}x = kx + C$ ($k$ is constant).

• $\int x^\alpha \mathrm{d}x = \frac{x^{\alpha + 1}}{\alpha + 1} + C$ ($\alpha \neq -1$).

• $\int \frac{1}{x}\mathrm{d}x = \ln|x| + C$.

• $\int \frac{1}{1 + x^2}\mathrm{d}x = \arctan x + C$.

• $\int \frac{1}{\sqrt{1 - x^2}}\mathrm{d}x = \arcsin x + C$.

• $\int \cos x \mathrm{d}x = \sin x + C$.

• $\int \sin x \mathrm{d}x = -\cos x + C$.

• $\int \frac{1}{\cos^2 x}\mathrm{d}x = \int \sec^2 x \mathrm{d}x = \tan x + C$.

• $\int \frac{1}{\sin^2 x}\mathrm{d}x = \int \csc^2 x \mathrm{d}x = -\cot x + C$.

• $\int \sec x \tan x \mathrm{d}x = \sec x + C$.

• $\int \csc x \cot x \mathrm{d}x = -\csc x + C$.

• $\int e^x \mathrm{d}x = e^x + C$.

• $\int a^x \mathrm{d}x = \frac{a^x}{\ln a} + C$ ($a > 0$ and $a \neq 1$)
  Remember to add +C

Properties of indefinite integrals # memorize

$$\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$
$$\int kf(x)dx=k\int f(x)dx$$

Substitution method

First type substitution

If $f(u)$ has an antiderivative and $u=\phi(x)$ is differentiable, then the substitution formula is
$$\int f[\phi(x)]\phi’(x)dx=\left[ \int f(u)du \right]_{u=\phi(x)}$$
Often use matching differentials.

Second type substitution

That is, use the first type in reverse.
Let $x=\phi(t)$ be monotonic and differentiable with $\phi’(t)\neq_{0}$. If $f[\phi(t)]\phi’(t)$ has an antiderivative, then
$$\int f(x)dx=\left[ \int f[\phi(t)]\phi’(t)dt \right]_{t=\phi^-1(x)}$$
where $t=\phi^-1(x)$ is the inverse function of $x=\phi(t)$.
When the integrand contains $\sqrt{ a^2\pm x^2 }$ or $\sqrt{ x^2-a^2 }$, use trigonometric substitution, letting $x=a\tan t$, $x=a\sin t$, $x=a\sec t$ to remove the radical. If the variable degree in the denominator is high, use reciprocal substitution $x=\frac{1}{t}$.

Integration by parts # memorize

$$\int uv’dx=uv-\int u’vdx$$
$$\int udv=uv-\int vdu$$

Integrals of rational functions

Algebraic preliminaries

A rational function is the ratio of two polynomials.
Use polynomial division to turn an improper fraction into a polynomial + proper fraction.

Integrals of rational functions

Definite integrals

Concepts and properties of definite integrals

Example

Definition of definite integral

A definite integral can be rigorously defined by the limit of Riemann sums. A common form is:

$$
\int_a^b f(x),dx=\lim_{n\to\infty}\sum_{i=1}^n f(x_i^*),\Delta x.
$$

A definite integral is a constant. It is 0 when $a=b$, and flips sign when $a>b$.

• If $f(x)$ is continuous on $[a,b]$, or bounded with only finitely many discontinuities, it is integrable.

• Geometric meaning: the area enclosed by $y=fx$, $x=a$, $x=b$, and the x-axis (negative if below the axis)

Properties of definite integrals

• Linearity: $\int_a^b\big(\alpha f(x)+\beta g(x)\big),dx=\alpha\int_a^b f(x),dx+\beta\int_a^b g(x),dx.$

• (Generalized) additivity of intervals: $\int_a^b f(x),dx=\int_a^c f(x),dx+\int_c^b f(x),dx.$

• Sign preservation: if $f(x)\geq0$ on $[a,b]$, then $\int_a^b f(x)\geq0$

• Order preservation: if $f(x)\geq g(x)$ on $[a,b]$, then $\int_a^b f(x)\geq\int_a^b g(x)$

• Absolute integrability: $|\int_a^b f(x)dx|\leq \int_a^b |f(x)|dx$

• Estimation theorem: if $f(x)$ has maximum $M$ and minimum $m$ on $[a,b]$, then $m(b-a)\leq \int_a^b f(x)\leq M(b-a)$

• Mean value theorem for integrals: $\int_a^b f(x)dx=f(\phi)(b-a)$

Fundamental theorem of calculus

Example

$\int_{a}^b f(x)dx=F(b)-F(a)$

Variable upper limit function

$\Phi(x)=\int_a^x f(t)dt, x \in[a,b]$
Is an antiderivative of $f(x)$ on $[a,b]$, and its derivative is
$\Phi’(x)=\frac{d}{dx} \int_{a}^x f(t)dt=f(x)$
That is, $F’(x)=f(x)$

Newton-Leibniz formula

If $F(x)$ is an antiderivative of continuous $f(x)$ on $[a,b]$, then
$\int_a^b f(x)dx = F(b)-F(a)$

Substitution and integration by parts for definite integrals

Substitution

First type substitution

If $f(x)$ is continuous on $[a,b]$, and $x=\phi (t)$ defined on an interval satisfies

• $\phi(t)$ has a continuous derivative on its domain

• $f(x)$ is continuous on the value range of $\phi(t)$

• $\phi(\alpha)=a, \phi(\beta)=b$
  then

• $\int_a^b f(x)dx=\int_\alpha^\beta f(\phi(t))\phi’(t)dt$
  that is

• $\int_\alpha^\beta f(x)dx=\int_a^b f(\phi(t))\phi’(t)dt$

Second type substitution

->properties of even/odd functions on symmetric intervals (double/0) / integral properties of periodic functions

Integration by parts

If $u(x)$ and $v(x)$ have continuous derivatives on $[a,b]$, then
$$\int_a^b u(x)v’(x)dx=u(x)v(x) |_{a}^{b} - \int_a^b u’(x)v(x)dx$$

Improper integrals

The integration interval is infinite or the integrand is unbounded.

Improper integrals over infinite intervals

Convergence and divergence, apply Newton-Leibniz.
(Convergent: unbounded horizontal graph has a finite value)

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Convergence of improper integrals over infinite intervals is equivalent to convergence of an antiderivative of the integrand at infinity.

Improper integrals with unbounded integrands

Convergence and divergence
(Convergent: unbounded vertical graph has a finite value)

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Applications of definite integrals

Geometric applications

Element method for definite integrals

1. Break big into small

2. Fix variable

3. Form a sum

4. Take a limit

   Area of plane figures

   Cartesian coordinates

   $$A=\int_a^b|f(x)|dx$$

   Polar coordinates

   $$A=\frac{1}{2}\int_\alpha^\beta[\phi(\Phi)]^2d\Phi$$

   Volume

   Volume of a solid with known cross-sectional area

   $$V=\int_a^bA(x)dx$$ ($A(x)$ is the base area)

   Volume of a solid of revolution

   $$V=\int_a^b \pi [f(x)]^2dx$$

   Arc length of a plane curve

• Can compute length

• Smooth curve

Physical applications of definite integrals

Ordinary differential equations

Basic concepts of ODEs

Concept of a differential equation

• An equation containing unknown derivatives or differentials is called a differential equation

• If the unknown is a function of one variable, it is an ordinary differential equation; multiple variables give a partial differential equation

• The order of the highest derivative is the order of the differential equation

Solutions of differential equations

Particular solution, general solution

First-order differential equations

Separable equations

That is, can be written as $g(y)dy=f(x)dx$
Separate variables: move $dx$ and $dy$ to different sides, then integrate both sides

Homogeneous equation

Form $\frac{dy}{dx}=f\left( \frac{y}{x} \right)$
Use variable substitution to convert to a separable equation, then solve

First-order linear differential equations

memorize

Homogeneous equation $\frac{dy}{dx}+P(x)y=0$ has general solution
$$y=Ce^{-\int P(x)dx}$$
Non-homogeneous equation $\frac{dy}{dx}+P(x)y=Q(x)$ has general solution:
$$y=e^{\int {Px(x)}}\left( \int Q(x)e^{-\int P(x)dx}+C \right)$$

Second-order linear differential equations

Structure of solutions for second-order homogeneous linear equations

Superposition principle

Structure of solutions for second-order non-homogeneous linear equations

$$y=Y+y*$$
$y*$ is a particular solution, and $Y$ is the general solution of the corresponding homogeneous equation

Structure of solutions for second-order constant-coefficient homogeneous equations

Characteristic equation

Structure of solutions for second-order constant-coefficient non-homogeneous equations