Separable Equations

A differential equation of the form $\frac{\mathrm{d} y }{\mathrm{d}x}=\frac{f(x)}{g(x)}$ is called separable because its variables can be separated. The solution to it is given by $\int g(y)\mathrm{d}y=\int f(x)\mathrm{d}x.$

Homogeneous Equations

A function of two variables, $f(x, y)$, is said to be homogeneous of degree $n$ if there is a constant $n$ such that $f(tx,ty)=t^nf(x,y)$ for all $t,x$ and $y$ for which both sides are defined. A differential equation of the form $M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y=0$ is homogeneous if $M$ and $N$ are both homogeneous functions of the same degree.

To solve this equation, let $y=xv$, $\mathrm{d}y=x\mathrm{d}v+v\mathrm{d}x$, then the equation will become separable. After solving for solution concerning $v$ and $x$, replace $v$ by $\frac{y}{x}$ and we can get the solution to the original equation.

Exact Equations

Given a function $f(x, y)$, its total differential, $\mathrm{d}f$, is defined as: $\mathrm{d}f=\frac{\partial f}{\partial x}\mathrm{d}x + \frac{\partial f}{\partial x}\mathrm{d}y.$ Then the family of curves $f(x, y) = c$ satisfies the differential equation, $\mathrm{d}f=\frac{\partial f}{\partial x}\mathrm{d}x + \frac{\partial f}{\partial x}\mathrm{d}y=0.$ So, if there exists a function $f(x, y)$ such that $M(x,y)=\frac{\partial f}{\partial x}$ and $N(x,y)=\frac{\partial f}{\partial y}$, then $M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y$ is called an exact differential and the equation $M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y=0$ is said to be an exact equation whose solution is the family $f(x,y)=c$.

If $\frac{\partial M}{\partial y}$ and $\frac{\partial N}{\partial x}$ are continuous, then the sufficient and necessary condition for the equation to be exact is that $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

Nonexact Equations And Itegreating Factors

If a nonexact equation has a solution, then an integrating factor $\mu(x,y)$ is guaranteed to exist such that $\mu(x,y)M(x,y)\mathrm{d}x+\mu(x,y)N(x,y)\mathrm{d}y=0$ is an exact equation.

If $\xi(x)=\frac{M_y-N_x}{N}$ is a function of $x$ alone, then $\mu(x)=e^{\int\xi(x)\mathrm{d}x}$.

If $\phi(y)=\frac{M_y-N_x}{-M}$ is a function of $y$ alone, then $\mu(x)=e^{\int\phi(y)\mathrm{d}y}$.

First-Order Linear Equations

A first-order linear equation is defined as a differential equation of the form: $\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x).$ The integrating factor for this equation is $\mu(x)=e^{\int P(x)\mathrm{d}x}$. Then $\frac{\mathrm{d}(\mu y)}{\mathrm{d}x}=\mu Q$. Integrating both sides gives the general solution $y=\frac{1}{\mu}\int \mu Q \mathrm{d}x.$

Higher-Order Linear Equations With constant Coefficients

The linear homogeneous differential equation of the
$n$th order with constant coefficients can be written as $y^{(n)}+ a_1 y^{(n-1)}+\cdots + a_n y=0,$
where $a_1,\ldots,a_n\in \mathbb{C}$. For each differential operator with constant coefficients, we can introduce the characteristic polynomial $L(\lambda)=\lambda^n+a_1\lambda^{n-1}+\cdots+a_n.$
The algebraic equation $L(\lambda)=0$ is called the characteristic equation of the differential equation.

If all roots of the characteristic equation are real and distinct, then the general solution is of the form $y=C_1e^{\lambda_1 x}+\cdots +C_ne^{\lambda_n x}$ where $\lambda_1,\ldots,\lambda_n\in \mathbb{R}$ are distinct roots and $C_1,\ldots,C_n\in\mathbb{C}$ are constants.
If all roots of the characteristic equation are real and multiple, then the general solution is of the form $y=\sum\limits_{i=1}^m\sum\limits_{j=1}^{k_i}C_{ij}x^{j-1}e^{\lambda_i x}$
where $\lambda_1,\ldots,\lambda_m\in \mathbb{R}$ are roots with multiplicity $k_1,\ldots,k_m$ and $\sum\limits_{i=1}^mk_i=n$, and $C_{ij}\in\mathbb{C}$ are constants.
If all roots of the characteristic equation are complex and distinct, then the general solution is of the form $y=\sum\limits_{j=1}^{\frac{n}{2}}e^{\alpha_j x}[C_{j1}\cos(\beta_j x)+C_{j2}\sin(\beta_j x)]$ where $\alpha_j\pm i\beta_j\in \mathbb{C}$ are distinct complex roots and $C_{j1},C_{j2}\in\mathbb{C}$ are constants.
If all roots of the characteristic equation are complex and multiple, then the general solution is of the form $y=\sum\limits_{j=1}^{\frac{n}{2}}\sum\limits_{l=1}^{k_j}e^{\alpha_j x}x^{l-1}[C_{j1}\cos(\beta_j x)+C_{j2}\sin(\beta_j x)]$ where $\alpha_j\pm i\beta_j\in \mathbb{C}$ are complex roots with multiplicity $k_j$ and $C_{j1},C_{j2}\in\mathbb{C}$ are constants.

Systems of Linear Differential Equations

In matrix form, a system of first-order linear homogeneous differential equations with constant coefficients is given by

$\mathbf{x}'(t)=\mathbf{A}\mathbf{x}(t),$

for $\mathbf{x}(t),\mathbf{x}’(t)\in\mathbb{R}$. In the case where $\mathbf{A}$ has $n$ linearly independent eigenvectors, this differential equation has the following general solution,

$\mathbf{x}(t)=c_{1}e^{\lambda _{1}t}\mathbf {u} _{1}+c_{2}e^{\lambda _{2}t}\mathbf {u} _{2}+\cdots +c_{n}e^{\lambda _{n}t}\mathbf{u}_{n},$

where $\lambda_1, \lambda_2, \ldots, \lambda_n$ are the eigenvalues of $\mathbf{A}$; $\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n$ are the respective eigenvectors of $\mathbf{A}$; and $c_1, c_2, \ldots, c_n$ are constants.