• #### The Integrating Factor Method

integrating factor method is a method of solving first order differential equations: It can be used for equations of the form $f(x)\frac{dy}{dx}+yg(x)=h(x)$ Divide by $f(x)$ to get $\frac{dy}{dx}+ \frac{yg(x)}{f(x)}= \frac{h(x)}{f(x)}$ Multiply by...

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Integration by parts is a commonly used technique of integration, used especially to integrate products. It is derived from the product rule for differentiating a product.. The product rule states that for functions $u, \; v$ , $\frac{(uv)}{dx}=... https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5349-integration-by-parts.html • #### Integrating With Factors How do you evaluate \[\int \frac{1}{1+e^x}dx$ ? Sometimes it is possible to multiple by a factor equal to 1 so that the integration becomes simpler. In this case we can multiply the integrand by $\frac{e^{-x}}{e^{-x}}$ to obtain $\int... https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5347-integrating-with-factors.html • #### Integration Using Partial Fractions We can integrate fractions where the denominator is a polynomial using partial fractions, which involves separating the fraction into simpler terms and integrating each one. Example: \[\int \frac{2}{x^2+3x+2} dx$ . We solve for $A$ and $B$ the...

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• #### Solving Differential Equations Using Perturbation Expansions

the solution to the unperturbed equation, Substitute this expression for into (2) to obtain This can be solved by the integrating factor method to obtain Substitution of the expressions for and into the third of these gives This can again be solved by...

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\ ]^t_0\] $\frac{dy}{dt}-v_0 + \frac{k}{m}y =-gt$ . Write this as $\frac{dy}{dt}+ \frac{k}{m}y =v_0-gt$ . Use the integrating factor method with integrating factor $e^{\frac{k}{m}t}$ . The equation becomes $e^{\frac{k}{m}t}... https://astarmathsandphysics.com/a-level-maths-notes/m4/5075-ball-thrown-upwards-subject-to-air-resistance-proportional-to-speed.html • #### Bernoullis Equation \frac{dt}{dx} +P(x)t=Q(x) \rightarrow \frac{dt}{dx}=P(x)t=Q(x)$ . The transformed equation can be solved by The Integrating Factor Method . Example: Solve $\frac{dy}{dx} - \frac{2}{x}y=xy^2$ > After the transformation $t=y^{1-2}=y^{-1}$ the...

https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5410-bernoullis-equation.html
• #### Differential Equatioins - Separating Variables

It is a very unusual thing to be given a differential equation that will just, well, integrate. Usually some manipulations must be performed, whether it is simplifying, grouping like terms, simplifying, making substitutions or separating variables –...

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• #### Simultaneous Differential Equations

(3) From (3) $f'=-g'$ then we can write (1) as $f''-2f'=4$ This last equation can bee solved using the integrating factor method. Multiply the equation throughout by $e^{\int{-2dx}}= e^{-2x}$ . We obtain $f' e^{-2x}-2f e^{-2x}=4e^{-2x}$ . We...

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• #### Separation of Variables

Separation of variables is a technique used to rearrange a first order differential equation into a form that can be integrated. $\frac{y}{x+2} \frac{dy}{dx}=e^y$ . Every factor containing $x$ is moved to the side contain $dy$ and all the...

https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5351-separation-of-variables.html
• #### Natural Logairthms

One number raised to the power of another is called a base. In the expression 3^4 the base is 3. The most common base is 10 – we count and measure things in multiples of 10 because we have 10 fingers on which to count. There is however, one base which...

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• #### Normalizing the Wavefunction

The Born interpretation gives the probability of finding a particle with wavefunction - I have shown the wavefunction here to be a function of here, though I need not have done and do not use this below - in the volume of space between and is The...

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• #### Solving a Differential Equation By Regrouping Terms

$(2x-y)dx+(2y-x)dx=0$ with the curve passing through $(2,1)$ cannot be solved by separation of variables, use of integrating factors, parts, substitution,... If we expand the brackets and rearrange however we get $2xdx+2ydy-(ydx+xdy)=0$ We can...

https://astarmathsandphysics.com/university-maths-notes/elementary-calculus/5402-solving-a-differential-equation-by-regrouping-terms.html
• #### Electric Displacement

Dielectric may not be neutral even when unpolarised. If the dielectric carries a charge density of free charges representing a net surplus or deficit of electrons in the atoms of the dielectric and is the charge density due to the polarizing effect of...

https://astarmathsandphysics.com/university-physics-notes/electricity-and-magnetism/1544-electric-displacement.html
• #### Properties of the Eigenfunctions

The eigenfunctions are the solutions of the eigenfunction equation the solutions for the one dimensional simple harmonic oscillator case, are polynomials in multiplied by a gaussian If the are normalized to unity they have the following properties: The...

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