Euler Lagrange Equation

Euler-Lagrange Equation for $\int F(x, y, y’) dx$: $$\frac{d}{dx} \left( \frac{\partial F}{\partial y’} \right) = \frac{\partial F}{\partial y}$$ – Dependent variable missing, E-L equation becomes: $$\frac{\partial F}{\partial y’} = constant$$ – Independent variable missing, E-L equation becomes: $$F – y’\frac{\partial F}{\partial y’} = constant$$ _____________ Several dependent variables: E.g. $F = F (x, y_1 , y_2 , … , y_n …

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Vector Analysis

4 vector operations Addition of two vectors Commutative: $\textbf{A + B = B + A}$ Associative: $\textbf{(A + B) + C = A + (B + C)}$ Multiplication by a scalar Distributive: $a(\textbf{A + B}) = a\textbf{A} + a\textbf{B}$, where a is a scalar Dot product of two vectors (Also known as scalar product) $\textbf{A . B} = \left|A\right|\left|B\right|cos \theta$ Commutative: …

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Basics Of Second Order Differential Equation

A second order differential equation is of the form $a \frac{d^{2}y}{dx^{2}} + b \frac{dy}{dx} + b \frac{dy}{dx} + cy = f(x)$. $f(x)$ is called a source term or forcing function. The differential equation is called a homogeneous equation IF $f(x) = 0$ non-homogeneous IF $f(x)$ is not 0. The steps involved in solving a homogeneous equation and non-homogeneous are quite …

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Variable Separable Differential Equation

A first order equation $\frac{dy}{dx} = f(x,y) $ is said to be variable separable if f(x,y) is of the form:$$ g(x)h(y) $$ Steps to solve $\frac{dy}{dx} = g(x)h(y) $: Swap the variables around: $\frac{1}{h(y)}dy = g(x) dx$ Evaluate: $\int \frac{1}{h(y)} \, dy = \int g(x) \, dx$ You’re done! (Note: you might want to attempt to solve for y explicitly)   Back …

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