A first order equation $\frac{dy}{dx} = f \left( x,y \right)$ is said to be a Bernoulli equation if f(x,y) is of the form $g \left( x \right) y + h \left( x \right) y^{a}$. Steps to solve $f \left( x,y \right) = g \left( x \right) y + h \left( x \right) y^{a}$: 1) Divide by ya to get $y^{-a} \frac{dy}{dx} …
$$\frac{dy}{dx} = p \left( x \right) y + q \left( x \right)$$ Procedure: – Calculate $I = \int p\left( x \right) \, dx$ – Find the integrating factors $e^{-I \left( x \right)}$ and $e^{I \left( x \right)}$ – Evaluate $\int e^{I \left( x \right)} q \left( x \right) \, dx$ – Hence the solution: $$y = e^{-I \left( x \right)} \left( …
Sub-Topics: – Steps For Solving First Order Differential Equation – First Order Linear Differential Equation – Bernoulli Differential Equation – Variable Separable Differential Equation – Homogeneous Differential Equation – Exact Differential Equation – Non-Exact Differential Equation Back To Mathematics For An Undergraduate Physics Course
Steps: 1) TRANSFORM the differential equation to the form: $$\frac{dy}{dx} = f \left( x,y \right)$$ 2) Test for Linear form: $$f \left( x,y \right) = g \left( x \right) y + h \left( x \right)$$ 3) Test for Bernoulli equation: $$f \left( x,y \right) = g \left( x \right) y + h \left( x \right) y^{a}$$ 4) Test for Variable …
There is a special integration “formula” that you can use to integrate products of Physics and engineering functions: eax sin bx cos cx ,where a, b and c are constants. $$\int f \, g = \frac{f\, g^{\prime}-f^{\prime}\, g}{u-v} + C$$ , where $f^{\prime \prime} = vf$ …
This post is meant to identify and briefly teach you the mathematics needed for a Physics course. (Not all) I will provide links to the websites that teach the relevant topics. Year 1 Calculus Integration without integration Differential Equations First Order Differential Equations Second Order Differential Equations Linear Algebra Coordinate Transformation Under Rotation Great lecture series by MIT (Youtube): httpvp://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 ____________________________________ …
Table Of Basic Integrals Basic Forms \begin{equation} \int x^n dx = \frac{1}{n+1}x^{n+1},\hspace{1ex}n\neq -1 \end{equation} \begin{equation} \int \frac{1}{x}dx = \ln |x| \end{equation} \begin{equation} \int u dv = uv – \int v du \end{equation} \begin{equation} \int \frac{1}{ax+b}dx = \frac{1}{a} \ln |ax + b| \end{equation} Integrals of Rational Functions \begin{equation} \int \frac{1}{(x+a)^2}dx = -\frac{1}{x+a} \end{equation} \begin{equation} \int (x+a)^n dx = \frac{(x+a)^{n+1}}{n+1}, n\ne -1 …