PHYS113 - Series and differential equations

One containing only full derivatives (only one independent variable)

One containing partial derivatives (multiple independent variables)

The highest order derivatives contained within the equation

The dependent variable or it's derivatives are not raised to a power other than 1.

An algebraic expression containing all arbitrary constants. It represents every possible solution

A solution which has had its arbitrary constants found. It satisifies the differential equation given the boundary conditions.

A function which satisfies the condition

$f(\lambda x_1, \lambda x_2, \lambda x_3...)=\lambda^n f(x_1,x_2,x_3...)$

It may also be considered by giving a weighting of one to each power ($x^2=2, \ y^{-4}=-4 \ etc$). The sum of weightings of each term in a homogenous equation must be constant throughout.

Any 1st order homogenous DE may be written in the form
$\frac{dy}{dx}=f\left(\frac{y}{x}\right)$

A new variable may be introduced: $v=\frac{y}{x}$

From this, it may be shown that
$\frac{dy}{dx}=\frac{dvx}{dx}=\frac{dv}{dx}x+v$

This gives rise to a seperable first order DE, which may be solved as usual

A differential equation in the form
$\frac{dy}{dx}+P(x)y=Q(x)$

By multiplying through by an integrating factor of $e^{\int P(x).dx}$. This gives an exact differential equation, with a solution in the form $y=\frac{\int IQ(x).dx}{I}$

An ODE in the form
$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0$

An ODE in the form
$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=f(x)$

If $y_1(x)$ and $y_2(x)$ are seperate solutions to a differential equation, then $y_1(x)+y_2(x)$ is also a solution

By solving the auxilliary equation $a\lambda^2+b\lambda+c=0$ and substituting in to the correct general solution

$y=Ae^{\lambda_1 x}+Be^{\lambda_2 x}$

$y=(A+Bx)e^{\lambda x}$

By solving the auxilliary equation to the complementary function (the homogenous equivalent to the left hand side) and substituting the solutions to this in to the general solution as for a homogenous 2nd order ODE. A trial solution is then used to find the solution to the inhomogenous component and these soltions added to those fro the complementary function. This gives a general solution.

An ordered list of elements $a_r$ which may be given in terms of the first term and the place in the sequence which the term occurs

A sequence given by
$\begin{matrix} a_0=t\\ a_{r+1}=a_r+s\\ \end{matrix} $
Where S is the constant diffeence between each term, and r

The sum of the terms in a sequence.

The sum of the first n terms of a sequence.

$S_N=\sum_{r=0}^{N-1}a_r=Na_0+\frac{N(N-1)}{2}s$
Where $a_0$ is the first term

A sequence given by
$a_r=a_0 q^r$
Where
$q=\frac{a_{r+1}{a_r}$
is the common ratio

An infinite series is convergent if its nth partial sum tends towards a consatnt value as n tends towards infinity.

$q\geq1$ - Divergent
$-1<q<1$ - Converges to $\frac{a_0}{1-q}$
$q\leq-1$- Diverges by oscillation

A series given by
$P(x)=\sum^{\infty}{r=0}c_r(x-a)^r$
Where $c_r$ and $a$ are constants.

By applying the ratio test