> [!NOTE] **Definition** (Solution of Scalar ODE) > Let $F(x,y, y'\dots,y^{(n)})=0$ be a [[Scalar Ordinary Differential Equation|scalar ordinary differential equation]]. A solution for $F$ is a [[Real Function|function]] $\phi: I \to \mathbb{R}$, for some [[Real intervals|interval]] $I \subset R,$ that is $n$-times [[Fréchet Differentiation|differentiable]] on $I$ for which $\forall x \in I, \quad F(x,\phi(x), \phi'(x),\dots,\phi^{(n)} (x)) = 0 .$ # Properties **Forms of a solution**: A solution is ***fully explicit*** iff the dependent variable is given as a combination of [[Elementary Function|elementary functions]] of the independent variable: for example, $y(t)=3\cos (5t)+8 \sin(t)$. A solution is **explicit** if $y$ is still given directly as a function of $t$, but as an expression involving an integral, for example $y(t)=1+\int_{0}^{t} e^{-s^{2}} \, ds.$ Here $y$ is still an explicit solution of $t$, but the integral cannot be evaluated in terms of elementary functions. An ***implicit*** form of the solution is an equation that relates the dependent and independent variables and involves no derivatives: e.g. $\ln y + 4\ln x - y-2x+4 = 0$ ([[@robinsonIntroductionOrdinaryDifferential2004|Robinson, 2004]]). **Existence and uniqueness of solution to IVPs:** 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. 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 $x_{0}$ and satisfies the Lipschitz condition 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 (Wikipedia). 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 of exact solutions**: | 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 $x'(t_{0})=v_{0}.$ | | System of Homogenous Linear First Order Ordinary Differential Equations with Real Coefficients: $\begin{align} x_{1}'(t)= a_{1,1}x_{1}(t)+ a_{1,2}x_{2}(t) \\ x_{2}'(t)= a_{2,1}x_{1} (t)+ a_{2,2}x_{2}(t) \end{align}$or equivalently, $\underline{x}'(t)=A\underline{x}(t)$ | By [[Solution to Homogenous Linear 2 x 2 System of First Order Ordinary Differential Equations with Real Coefficients]], | | | | | |