From 28f09ddee435d8614bfa5d8a9d61d5a5f2aecc17 Mon Sep 17 00:00:00 2001 From: Luc Date: Wed, 25 Oct 2023 22:27:17 +0200 Subject: [PATCH] Added and updated ODE section. --- config/en/mkdocs.yaml | 1 + .../laplace-transform.md | 1 + .../second-order-ode.md | 8 +- .../systems-of-linear-ode.md | 73 +++++++++++++++++++ 4 files changed, 79 insertions(+), 4 deletions(-) create mode 100644 docs/en/mathematics/ordinary-differential-equations/laplace-transform.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 7b48f1f..a5a08b7 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -58,6 +58,7 @@ nav: - 'First order differential equations': mathematics/ordinary-differential-equations/first-order-ode.md - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md + - 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md - 'Physics': - 'Start': physics/start.md diff --git a/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md b/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md new file mode 100644 index 0000000..ccf1e86 --- /dev/null +++ b/docs/en/mathematics/ordinary-differential-equations/laplace-transform.md @@ -0,0 +1 @@ +# The Laplace transform \ No newline at end of file diff --git a/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md index 8828373..ca8a37d 100644 --- a/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md +++ b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md @@ -40,7 +40,7 @@ $$ L[y(t)] = \lambda^2 e^{\lambda t} + p \lambda e^{\lambda t} + q e^{\lambda t} = e^{\lambda t} (\lambda^2 + p \lambda + q) = 0, $$ -obtaining the characteristic equation $\Chi(\lambda) = \lambda^2 + p \lambda + q = 0$. If two roots $\lambda_1,\lambda_2 \in \mathbb{C}$ are found the solution space is +obtaining the characteristic equation $\chi(\lambda) = \lambda^2 + p \lambda + q = 0$. If two roots $\lambda_1,\lambda_2 \in \mathbb{C}$ are found the solution space is $$ y(t) = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t}, \quad c_1,c_2 \in \mathbb{C}, @@ -56,7 +56,7 @@ $$ #### Example -Let the homogeneous linear second-order ode be given by $\ddot y + 4 \dot y + 8y = 0$. Then the characteristic equation is given by $\Chi(\lambda) = \lambda^2 + 4\lambda + 8 = 0$ with solutions $\lambda_1 = -2 + 2i$ and $\lambda_2 = -2 - 2i$. Then the general solution is given by +Let the homogeneous linear second-order ode be given by $\ddot y + 4 \dot y + 8y = 0$. Then the characteristic equation is given by $\chi(\lambda) = \lambda^2 + 4\lambda + 8 = 0$ with solutions $\lambda_1 = -2 + 2i$ and $\lambda_2 = -2 - 2i$. Then the general solution is given by $$ y(t) = c_1 e^{(-2 + 2i)1 t} + c_2 e^{(-2 - 2i) t}, \quad c_1,c_2 \in \mathbb{C}, @@ -118,7 +118,7 @@ named the Wronskian and we can solve for $c_1(t)$ and $c_2(t)$ by integration. #### Ansatz method -Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \R$ +Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \mathbb{R}$ | Inhomogeneity | Particular solution | | ------ | --------------- | @@ -129,5 +129,5 @@ Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhom | $L[y] = P_n e^{\alpha t} \cos \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | | $L[y] = P_n e^{\alpha t} \sin \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | -Choose $m \in \N \cup \{0\}$ as small as possible such that no term in the ansatz solves the homogeneous equation $L[y] = 0$. +Choose $m \in \mathbb{N} \cup \{0\}$ as small as possible such that no term in the ansatz solves the homogeneous equation $L[y] = 0$. diff --git a/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md index b31ad71..1c56cb2 100644 --- a/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md +++ b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md @@ -1,2 +1,75 @@ # Systems of linear ordinary differential equations +## Homogeneous systems of linear ODEs with constant coefficients + +Let $\mathbb{K} = \mathbb{R} \lor \mathbb{C}$, $n \in \mathbb{N}$ and $A \in \mathbb{R}^{n \times n}$. Seek differentiable functions $y:\mathbb{R} \to \mathbb{K}^n$ such that + +$$ +\mathbf{\dot y(t)} = A \mathbf{y}(t), \qquad t \in \mathbb{R} +$$ + +The solutions from a linear space, therefore the general solutions can be written as, + +$$ +\mathbf{\dot y(t)} = \sum_{k=1}^n c_k \mathbf{y}_k(t), \qquad c_k \in \mathbb{K} +$$ + +where $\{\mathbf{y_1}, \dots, \mathbf{y_n}\}$ is a linear independent set of solutions, i.e. the basis of the solutions space. + +Assume now that $A$ is diagonalizable, and let $\{\mathbf{v_1}, \dots, \mathbf{v_n}\}$ be a basis of $\mathbb{K}^n$ consisting of eigenvectors of A. + +$$ +AV = VD, \qquad \text{with } D = \begin{pmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \end{pmatrix} +$$ + +then $A = VDV^{-1}$, let $\mathbf{z}(t) = V^{-1} \mathbf{y}(t)$ + +$$ +\begin{array}{ll} +&\mathbf{\dot z} = V^{-1} \mathbf{\dot y} = V^{-1} A \mathbf{y} = V^{-1} V D V^{-1} = D \mathbf{z}, \\ +& \mathbf{\dot z} = D \mathbf{z} \implies \mathbf{z}(t) = \mathbf{c} e^{\lambda t}. +\end{array} +$$ + +Obtaining the general solution + +$$\mathbf{y}(t) = V \mathbf{z}(t) = \sum_{k=1}^n c_k \mathbf{v_k} e^{\lambda_k t}. +$$ + +## Inhomogeneous systems of linear ODEs with constant coefficients + +Let $I \subseteq \mathbb{R}$ be an interval, $\mathbf{f}: I \to \mathbb{R}$ continuous. Find functions $\mathbf{y}: I \to \mathbb{R}^n$ such that + +$$ +\mathbf{\dot y}(t) = a \mathbf{y}(t) + \mathbf{f}(t), \qquad t \in I. \qquad (*) +$$ + +*Theorem*: let $\mathbf{y}_p: I \to \mathbb{R}^n$ a particular solution for $(*)$ and $\mathbf{y}_H$ the general solution to the homegeneous system. Then the general solutions of the inhomogeneous system $(*)$ is given by + +$$ +\mathbf{y}(t) = \mathbf{y}_p(t) + \mathbf{y}_H(t), \qquad t \in I +$$ + +*Proof*: similar to 1d case, will possibly be added later. + +### Method of variation of parameters + +Let $\{\mathbf{y_1}, \dotsc, \mathbf{y_n}\}$ be a basis for the solution space of the homogeneous system. Ansatz: + +$$ +\mathbf{y}_p(t) = \sum_{k=1}^n c_k(t) \mathbf{y}_k(t) = (\mathbf{y}_1, \dots, \mathbf{y}_n) \begin{pmatrix} c_1(t) \\ \vdots \\ c_n(t) \end{pmatrix} = Y(t) \mathbf{c}(t), +$$ + +where $c_1(t), \dots, c_n(t): I \to \mathbb{R}$ are to be determined. + +Then: + +$$ +\begin{align*} +\mathbf{\dot y}_p &= \sum_{k=1}^n \dot c_k(t) \mathbf{y}_k(t) + \sum_{k=1}^n c_k(t) \mathbf{\dot y}_k(t), \\ + &= \sum_{k=1}^n \dot c_k(t) \mathbf{y}_k(t) + A \sum_{k=1}^n c_k(t) \mathbf{y}_k(t), \\ + &= Y(t) \mathbf{\dot c}(t) + A \mathbf{y}_p(t). +\end{align*} +$$ + +Demanding that: $Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)$. Then $\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)$ is nonsingular. Then solve for $\mathbf{c}(t)$. \ No newline at end of file