# First-order Ordinary Differential Equations cheatsheet Star

## Introduction

Differential Equations A differential equation is an equation containing derivatives of a dependent variable $y$ with respect to independent variables $x$. In particular,

- Ordinary Differential Equations (ODE) are differential equations having one independent variable.

- Partial Differential Equations (PDE) are differential equations having two or more independent variables.

Order An ODE is said to be of order $n$ if the highest derivative of the unknown function in the equation is the $n^{th}$ derivative with respect to the independent variable.

Linearity An ODE is said to be linear only if the function $y$ and all of its derivatives appear by themselves. Thus, it is of the form:

$\boxed{a_n(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+...+a_1(x)y'+a_0(x)y+b(x)=0}$

## Direction Field Method

Implicit form The implicit form of an ODE is where $y'$ is not separated from the remaining terms of the ODE. It is of the form:

$\boxed{F(x,y,y')=0}$

Remark: Sometimes, $y'$ cannot be separated from the other terms and the implicit form is the only one that we can write.

Explicit form The explicit form of an ODE is where $y'$ is separated from the remaining terms of the ODE. It is of the form:

$\boxed{y'=f(x,y)}$

Direction field method The direction field method is a graphical representation for the solution of ODE $y'=f(x,y)$ without actually solving for $y(x)$. Here is the procedure:

1. Determine the values $(x_i,y_i)$ that form the grid.

2. Compute the slope $f(x_i,y_i)$ for each point of the grid.

3. Report the associated vector for each point of the grid.

## Separation of Variables

Separable An ODE is said to be separable if it can be written in the form:

$\boxed{f(x,y)=g(x)h(y)}$

Reduction to separable form The following table sums up the variable changes that allow us to change the ODE $y'=f(x,y)$ to $u'=g(x,u)$ that is separable.

 Original form Change of variables New form $y'=f\left(\frac{y}{x}\right)$ $u\triangleq\frac{y}{x}$ $u'x+u=f(u)$ $y'=f\left(ax+by+c\right)$ $u\triangleq ax+by+c$ $\frac{u'-a}{b}=f(u)$

## Equilibrium

Characterization In order for an ODE to have equilibrium solutions, it must be (1) autonomous and (2) have a value $y^*$ that makes the derivative equal to 0, i.e:

$(1)\quad\boxed{\frac{dy}{dt}=f(\require{cancel} \xcancel{t},y)=f(y)}\quad\textrm{ and }\quad(2)\quad\boxed{\exists\textrm{ }y^*, \frac{dy^*}{dt} = f(y^*)=0}$

Stability Equilibrium solutions can be classified into 3 categories:

- Unstable: solutions run away with any small change to the initial conditions.
- Stable: any small perturbation leads the solutions back to that solution.
- Semi-stable: a small perturbation is stable on one side and unstable on the other.

## Linear first-order ODE technique

Standard form The standard form of a first-order linear ODE is expressed with $p(x), r(x)$ known functions of $x$, such that:

$\boxed{y'+p(x)y=r(x)}$

Remark: If $r=0$, then the ODE is homogenous, and if $r\neq0$, then the ODE is inhomogeneous.

General solution The general solution $y$ of the standard form can be decomposed into a homogenous part $y_h$ and a particular part $y_p$ and is expressed in terms of $p(x), r(x)$ such that:

$\boxed{y=y_h+y_p}\quad\quad\textrm{with}\quad\quad\boxed{y_h=Ce^{-\int pdx}}\quad\textrm{and}\quad\boxed{y_p=e^{-\int pdx}\times\int\left[r e^{\int pdx}\right]dx}$

Remark 1: Here, for any function $p$, the notation $\displaystyle\int pdx$ denotes the primitive of $p$ without additive constant.

Remark 2: The term $e^{-\int pdx}$ is called the basis of the ODE and $e^{\int pdx}$ is called the integrating factor.

Reduction to linear form The one-line table below sums up the change of variables that we apply in order to have a linear form:

 Name, setting Original form Change of variables New form Bernoulli, $n\in\mathbb{R}\backslash\{0,1\}$ $y'+p(x)y=q(x)y^n$ $u\triangleq y^{1-n}$ $u'+(1-n)p(x)u=(1-n)q(x)$

## Existence and uniqueness of an ODE

Here, we are given an ODE $y'=f(x,y)$ with initial conditions $y(x_0)=y_0$.

Existence theorem If $f(x,y)$ is continuous at all points in a rectangular region containing $(x_0,y_0)$, then $y'=f(x,y)$ has at least one solution $y(x)$ passing through $(x_0,y_0)$.

Remark: If the condition does not apply, then we cannot say anything about existence.

Uniqueness theorem If both $f(x,y)$ and $\frac{\partial f}{\partial y}(x,y)$ are continuous at all points in a rectangular region containing $(x_0,y_0)$, then $y'=f(x,y)$ has a unique solution $y(x)$ passing through $(x_0,y_0)$.

Remark: If the condition does not apply, then we cannot say anything about uniqueness.

## Numerical methods for ODE - Initial value problems

In this section, we would like to find $y(t)$ for the interval $[0,t_f]$ that we divide into $N+1$ equally-spaced points $t_0< t_1 < ... < t_N = t_f$, such that:

$\frac{dy}{dt}=f(t,y)\quad\textrm{with}\quad y(0)=y_0$

Error In order to assess the accuracy of a numerical method, we define its local and global errors $\epsilon_{\textrm{local}}, \epsilon_{\textrm{global}}$ as follows:

$\boxed{\epsilon_{\textrm{local}}=|y^{\textrm{exact}}(t_n)-y^{\textrm{numerical}}(t_n)|}\quad\textrm{and}\quad\boxed{\epsilon_{\textrm{global}}=\sqrt{\frac{\displaystyle\sum_{n=1}^N|y^{\textrm{exact}}(t_n)-y^{\textrm{numerical}}(t_n)|^2}{N}}}$

Remark 1: If $\epsilon_{\textrm{local}}=O(h^k)$, then $\epsilon_{\textrm{global}}=O(h^{k-1})$.

Remark 2: When we talk about the 'error' of a method, we refer to its global error.

Taylor series The Taylor series giving the exact expression of $y_{n+1}$ in terms of $y_n$ and its derivatives is:

$\boxed{y_{n+1}=y_n+hy_n'+\frac{h^2}{2}y_n''+\frac{h^3}{6}y_n'''+...=\sum_{k=0}^{+\infty}\frac{h^k}{k!}y_n^{(k)}}$

We can also have an expression of $y_n$ in terms of $y_{n+1}$ and its derivatives:

$\boxed{y_{n}=y_{n+1}-hy_{n+1}'+\frac{h^2}{2}y_{n+1}''-\frac{h^3}{6}y_{n+1}'''+...=\sum_{k=0}^{+\infty}\frac{(-h)^k}{k!}y_{n+1}^{(k)}}$

Stability The stability analysis of any ODE solver algorithm is performed on the model problem, defined by:

$\boxed{y'=\lambda y\quad\textrm{ with }\quad y(0)=y_0\quad\textrm{and}\quad\lambda < 0}$

which gives $y_n=y_0\sigma^n$, for which $h$ verifies the condition $|\sigma(h)| < 1$.

Euler methods The Euler methods are numerical methods that aim at estimating the solution of an ODE:

 Type Update formula Error Stability condition Forward Euler $y_{n+1}=y_n+hf(t_n,y_n)$ $O(h)$ $h<\frac{2}{|\lambda|}$ Backward Euler $y_{n+1}=y_n+hf(t_{n+1},y_{n+1})$ $O(h)$ None

Runge-Kutta methods The table below sums up the most commonly used Runge-Kutta methods:

 Method Update formula Error Stability condition RK1 Euler's $y_{n+1}=y_n+hk_1$where $k_1=f(t_n,y_n)$ $O(h)$ $\displaystyle h<\frac{2}{|\lambda|}$ RK2 Heun's $y_{n+1}=y_n+h\left(\frac{1}{2}k_1+\frac{1}{2}k_2\right)$where $k_1=f(t_n,y_n)$and $k_2=f(t_n+h,y_n+hk_1)$ $O(h^2)$ $\displaystyle h<\frac{2}{|\lambda|}$ RK3 Kutta's $y_{n+1}=y_n+h\left(\frac{1}{6}k_1+\frac{2}{3}k_2+\frac{1}{6}k_3\right)$where $k_1=f(t_n,y_n)$and $k_2=f(t_n+\frac{1}{2}h,y_n+\frac{1}{2}hk_1)$and $k_3=f(t_n+h,y_n-hk_1+2hk_2)$ $O(h^3)$ $\displaystyle h<\frac{2.5}{|\lambda|}$ RK4 Classic $y_{n+1}=y_n+h\left(\frac{1}{6}k_1+\frac{1}{3}k_2+\frac{1}{3}k_3+\frac{1}{6}k_4\right)$where $k_1=f(t_n,y_n)$and $k_2=f(t_n+\frac{1}{2}h,y_n+\frac{1}{2}hk_1)$and $k_3=f(t_n+\frac{1}{2}h,y_n+\frac{1}{2}hk_2)$and $k_4=f(t_n+h,y_n+hk_3)$ $O(h^4)$ $\displaystyle h<\frac{2.8}{|\lambda|}$

## System of Linear ODEs

Definition A system of $n$ first order linear ODEs

$\begin{cases} y_1'=a_{11}y_1+...+a_{1n}y_n\\ \vdots\\ y_n'=a_{n1}y_1+...+a_{nn}y_n\\ \end{cases}$

can be written in matrix form as:

$\boxed{\vec{y}'=A\vec{y}}$

where $A=\left(\begin{array}{ccc}a_{11}& \cdots& a_{1n}\\\vdots& \ddots & \vdots\\a_{n1}& \cdots& a_{nn}\end{array}\right)$ and $\vec{y}=\left(\begin{array}{c}y_1\\\vdots\\y_n\end{array}\right)$

Characteristic equation The characteristic equation of a linear system of $n$ equations represented by $A$ is given by:

$\boxed{\textrm{det}(A-\lambda I)=0}$

For $n=2$, this equation can be written as:

$\boxed{\lambda^2-(a_{11}+a_{22})\lambda+(a_{11}a_{22}-a_{12}a_{21})=0}$

Eigenvector, eigenvalue The roots $\lambda$ of the characteristic equation are the eigenvalues of $A$. The solutions $\vec{v}$ of the equation $A\vec{v}=\lambda I$ are called the eigenvectors associated with the eigenvalue $\lambda$.

System of homogeneous ODEs The resolution of the system of 2 homogeneous linear ODEs $\vec{y}'=A\vec{y}$ is detailed in the following table:

 Case Eigenvalues $\leftrightarrow$ Eigenvectors Solution Real distinct eigenvalues $\lambda_1\leftrightarrow\vec{\eta}_{\lambda_1}$ $\lambda_2\leftrightarrow\vec{\eta}_{\lambda_2}$ $\vec{y}=C_1\vec{\eta}_{\lambda_1}e^{\lambda_1t}+C_2\vec{\eta}_{\lambda_2}e^{\lambda_2t}$ Double root eigenvalues $\lambda\leftrightarrow\vec{\eta}$ $\vec{\rho}\textrm{ s.t. }(A-\lambda I)\vec{\rho}=\vec{\eta}$ $\vec{y}=[(C_1+C_2t)\vec{\eta}+C_2\vec{\rho}]e^{\lambda t}$ Complex conjugate eigenvalues $\alpha+i\beta\leftrightarrow\vec{\eta}_{R}+i\vec{\eta}_{I}$ $\alpha-i\beta\leftrightarrow\vec{\eta}_{R}-i\vec{\eta}_{I}$ $\vec{y}=C_1(\cos(\beta t)\vec{\eta}_R-\sin(\beta t)\vec{\eta}_I)e^{\alpha t}$ $\quad+ C_2(\cos(\beta t)\vec{\eta}_I+\sin(\beta t)\vec{\eta}_R)e^{\alpha t}$