> [!NOTE] Definition (Scalar ODE) > Let $n\in \mathbb{N}^{+}.$ A scalar (or one-dimensional) ordinary differential equation of [[Order of a Differential Equation|order]] $n$ is an [[Ordinary Differential Equation|ODE]] of the form $F(x,y,y',\dots,y^{(n)}) = 0 \tag{1}$for some function $F: (\alpha, \beta)\times \mathbb{R}^{n+1} \to \mathbb{R}$ with a given interval $(\alpha,\beta) \subset \mathbb{R}.$ # Properties **Classification**: The equation $(1)$ is [[Autonomous Scalar Ordinary Differential Equation|autonomous]] iff the independent variable solely occurs as an argument of the dependent variable. The equation is [[Homogeneous Scalar Ordinary Differential Equation|homogenous]] iff $F(t,0,0,\dots,0) = 0$ for all $t\in(\alpha,\beta).$ The equation is [[Linear Scalar Ordinary Differential Equation|linear]] iff the all the dependent variables and their derivatives appear to the first power. **Exact solution**: An [[Solution to Scalar Ordinary Differential Equation|exact solution]] of an ODE is a function that satisfies the ODE. A fully **explicit** form of the solution is an expression for the dependent variable that solely involves elementary functions of the independent variable. An **implicit** form of the solution is an equation that involves elementary functions of the dependent and independent variables and involves no derivatives, e.g. $\ln y + 4\ln x -y -2x+4=0.$ An [[Initial Value Problem for Scalar Ordinary Differential Equation|initial value problem]] is a differential equation together some initial condition. A solution of an IVP is a solution of the ODE that satisfies the initial condition. **Existence and uniqueness of solution to IVPs:** There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The [[Picard–Lindelöf theorem|Picard–Lindelöf theorem]] guarantees a unique solution to $x'(t)=f(t,x(t))$ with $x(t_{0})=x_{0}$ on some interval containing $t_{0}$ if $f$ is continuous on a region containing $t_{0}$ and $y_{0}$ and satisfies the L on the variable $x.$ In its basic the theorem only guarantees a local result, though the latter can be extended to give a global result, for example, if the conditions of [[Uniqueness Theorem for Explicit First Order Initial Value Problem|Grönwall's inequality]] are met. An [[Example of Initial Value Problem with Infinitely Many Solutions|example of an initial-value problem with a non-unique solution]] is: $x'(t)=\sqrt{ x(t) },\quad x(0)=0.$ # Applications **Examples**: | ODE | Form of every solution | IVP Solution (solution that satisfies $x(t_{0})=x_{0}$) | | | ----------------------------------------------------------------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Antiderivative]]: $\frac{d}{dt} x(t) = f(t), \quad t\in (\alpha,\beta).$Note that the above equation may be written as $x(t)= \int f(t) \, dt $ | By [[Fundamental theorem of calculus]], $x(t)= c+ \int_{\alpha}^{t} f(\tau) \, d\tau$that is, any two antiderivatives differ by at most a constant. | By [[Solution to Trivial Differential Equation Initial Value Problem]], $x(t)=x_{0}+ \int_{t_{0}}^{t} f(\tau) \, d\tau$ | | | [[Separable Differential Equation | Separable Equation]]: $\frac{d}{dt} x(t)= f(x)g(t)$ | By [[Implicit Solution to Separable Differential Equation]], $\int \frac{1}{f(x)} \, dx = \int g(t) \, dt $ | By [[Implicit Solution to Initial Value Problem for Separable Equation]], $\int_{x_{0}}^{x(t)} \frac{1}{f(\hat{x})} \, d\hat{x} = \int_{t_{0}}^{t} g(\tau)\, d\tau $ | | [[First Order Linear Ordinary Differential Equation]]: $\frac{d} {dt} x(t)+p(t)x(t) = q(t)$ | By [[Solution to First Order Linear Ordinary Differential Equation]], $x(t) e^{P(t)} = \int q(t) e^{P(t)} \, dt $ | By [[Implicit Solution to First Order Linear Ordinary Differential Equation Initial Value Problem]],$x(t)e^{P(t)}= x_{0} + \int_{t_{0}}^{t} q(\tilde{t})e^{P(\tilde{t})} \, d\tilde{t} $where $P(t)=\int p(t) \, dt.$ | | | [[Bernoulli Equation\|Bernoulli Equations]]: $\frac{d}{dt}x(t)+p(t)x(t)=q(t)x^{n}(t)$for $n=2,3,4\dots$ | By [[Solution to Bernoulli Equation]], $\begin{align}&\frac{d}{dt}u(t)+(1-n)p(t)u(t) \\ &=(1-n)q(t) \end{align}$ where $u(t)=x^{1-n}(t).$ | Same as above. | | | | | | | | | | | | | [[Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]]: $a \frac{d^{2}}{dt^{2}}x(t) +bx(t) + cx(t) = s(t)$ | By [[Solution to Inhomogeneous Second Order Linear Scalar Ordinary Differential Equation with Real Coefficients]], if $s(t)=0$ for all $t$ the form of all solutions is $x(t) = \begin{cases} l_{1}e^{\lambda_{1}t} + l_{2} e^{\lambda_{2}t} & b^{2}-4ac > 0, \\l_{1}e^{\lambda t} + l_{2}te^{\lambda t} & b^{2}-4ac=0 \;(\text{where }\lambda:= \lambda_{1}=\lambda_{2}), \\e^{p t} (l_1 \sin q t + l_2 \cos q t) & b^{2}-4ac<0 \; (\text{where } \lambda_{1,2}=p \pm i\,q ) \end{cases}$otherwise $x(t)=x_{p}(t)+x_{c}(t)$ where $x_{p}$ is the 'guessed solution' and $x_c(t)$ is the solution to the homogenous equation obtained by setting $s=0.$ | Find $l_{1},l_{2}\in \mathbb{R}$ by solving linear system obtained by substituting initial conditions: $x(t_{0})=x_{0}$ and $x'(t_{0})=v_{0}.$ | | Analysis: [[Scaling Procedure for Scalar Ordinary Differential Equations]].