First-order Ordinary Differential Equations cheatsheet

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By Afshine Amidi and Shervine Amidi

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:

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:

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:

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:

Separation of Variables

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

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.

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:

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:

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:

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:

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:

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:

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:

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

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

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:

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

System of Linear ODEs

Definition A system of $n$ first order linear ODEs

can be written in matrix form as:

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:

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

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: