First and Second Order Differential Equations

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nivedh_89

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Table of Trigonometric Identities

Reciprocal identities

Pythagorean Identities

Quotient Identities

Co-Function Identities

Even-Odd Identities

Sum-Difference Formulas

Double Angle Formulas

Power-Reducing/Half Angle Formulas

Sum-to-Product Formulas

Product-to-Sum Formulas

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`0`	`0`	`0`	`1`	`0`


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First and Second Order Differential Equations

First Order Differential equations

A first order differential equation is of the form:

Linear Equations:

The general general solution is given by

where

is called the integrating factor.

Separable Equations:

(1): Solve the equation g(y) = 0 which gives the constant solutions.
(2): The non-constant solutions are given by

Bernoulli Equations:

(1): Consider the new function .
(2): The new equation satisfied by v is
(3): Solve the new linear equation to find v.
(4): Back to the old function y through the substitution .
(5): If n > 1, add the solution y=0 to the ones you got in (4).

Homogenous Equations:

is homogeneous if the function f(x,y) is homogeneous, that is

By substitution, we consider the new function

The new differential equation satisfied by z is

which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by

Do not forget to go back to the old function y = xz.

Exact Equations:

is exact if

The condition of exactness insures the existence of a function F(x,y) such that

All the solutions are given by the implicit equation

Second Order Differential equations

Homogeneous Linear Equations with constant coefficients:

Write down the characteristic equation

(1): If and are distinct real numbers (this happens if ), then the general solution is
(2): If (which happens if ), then the general solution is
(3): If and are complex numbers (which happens if ), then the general solution is

where

that is

Non Homogeneous Linear Equations:

The general solution is given by

where

is a particular solution and

is the general solution of the associated homogeneous equation

In order to find

two major techniques were developed.

: Method of undetermined coefficients or Guessing Method
This method works for the equation

where a, b, and c are constant and

where is a polynomial function with degree n. In this case, we have

where

The constants and have to be determined. The power s is equal to 0 if is not a root of the characteristic equation. If is a simple root, then s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x) satisfies the following

where are of the forms cited above, then we split the original equation into N equations

then find a particular solution . A particular solution to the original equation is given by
: Method of Variation of Parameters
This method works as long as we know two linearly independent solutions of the homogeneous equation

Note that this method works regardless if the coefficients are constant or not. a particular solution as

where and are functions. From this, the method got its name.
The functions and are solutions to the system:

which implies

Therefore, we have

Euler-Cauchy Equations:

where b and c are constant numbers. By substitution, set

then the new equation satisfied by y(t) is

which is a second order differential equation with constant coefficients.

(1): Write down the characteristic equation
(2): If the roots and are distinct real numbers, then the general solution is given by
(2): If the roots and are equal ( ), then the general solution is
(3): If the roots and are complex numbers, then the general solution is

where and .

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DEFINITE INTEGRALS CONTAINING HYPERBOLIC FUNCTIONS

1.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin ax}{\sinh bx}dx=\displaystyle \frac{\pi}{2b}\tanh\displaystyle \frac{a\pi}{2b}$
2.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos ax}{\cosh bx}dx=\displaystyle \frac{\pi}{2b}\displaystyle \frac{1}{\cosh (a\pi/2b)}$
3.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{x dx}{\sinh ax}=\displaystyle \frac{\pi^2}{4a^2}$
4.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{x^n dx}{\sinh ax}=\displaysty... ...tyle \frac{1}{2^{n+1}}+\displaystyle \frac{1}{3^{n+1}}+\cdot\cdot\cdot \right\}$
5.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}+1}dx=\displaystyle \frac{\pi}{2b}\csc\displaystyle \frac{a\pi}{b}-\displaystyle \frac{1}{2a}$
6.: $\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sinh ax}{e^{bx}-1}dx=\displaystyle \frac{1}{2a}-\displaystyle \frac{\pi}{2b}\cot\displaystyle \frac{a\pi}{b}$

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DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS

Note that all the constant are positive.

1.: $displaystyleint_{0}^{pi}sin mx sin nx dx=left{ egin{array}{ll} displa... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight.$
2.: $displaystyleint_{0}^{pi} cos mx cos nx dx=left{ egin{array}{ll} displ... ...in} &mbox{if $m$space and $n$space integers and};; m=n end{array} ight.$
3.: $displaystyleint_{0}^{pi}sin mx cos nx dx=left{ egin{array}{ll} displa... ... $m$space and $n$space integers and $m+n$space even} \ end{array} ight.$
4.: $displaystyleint_{0}^{pi/2} sin^2 x dx=int_{0}^{pi/2}cos^2 dx=displaystyle rac{pi}{4}$
5.: $displaystyleint_{0}^{pi/2}sin^{2m} x dx=int_{0}^{pi/2}cos^{2m} x dx=dis... ...cdot 4cdot 6cdot cdotcdotcdot 2m}left(displaystyle rac{pi}{2} ight)$ ,
m=1,2,...
6.: $displaystyleint_{0}^{pi/2}sin^{2m+1}x dx=int_{0}^{pi/2}cos^{2m+1}x dx=d... ...c{2cdot 4cdot 6cdotcdotcdot 2m}{1cdot 3cdot 5cdot cdotcdotcdot 2m+1}$ ,
m=1,2,...
7.: $displaystyleint_{0}^{pi/2}sin^{2p-1}x cos^{2q-1}x dx=displaystyle rac{Gamma(p)Gamma(q)}{2Gamma(p+q)}$
8.: $displaystyleint_{0}^{infty}displaystyle rac{sin px}{x}dx=left{ egin{... ...n} p>0 \ 0&hspace{.3in} p=0 \ -pi/2&hspace{.3in} p<0 end{array} ight.$
9.: $displaystyleint_{0}^{infty}displaystyle rac{sin pxcos qx}{x}dx=left{ ... ...0\ pi/2&hspace{.3in} 0<p<q\ pi/4&hspace{.3in} p=q>0 end{array} ight.$
10.: $displaystyleint_{0}^{infty}displaystyle rac{sin px sin qx}{x^2}dx=left... ...hspace{.3in} 0<pleq q \ pi q/2&hspace{.3in} pgeq q>0 end{array} ight.$
11.: $displaystyleint_{0}^{infty}displaystyle rac{sin^2 px}{x^2}dx=displaystyle rac{pi p}{2}$
12.: $displaystyleint_{0}^{infty}displaystyle rac{1-cos px}{x^2}dx=displaystyle rac{pi p}{2}$
13.: $displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x}dx=lndisplaystyle rac{q}{p}$
14.: $displaystyleint_{0}^{infty}displaystyle rac{cos px-cos qx}{x^2}dx=displaystyle rac{pi(q-p)}{2}$
15.: $displaystyleint_{0}^{infty}displaystyle rac{cos mx}{x^2+a^2}dx=displaystyle rac{pi}{2a}e^{-ma}$
16.: $displaystyleint_{0}^{infty}displaystyle rac{xsin mx}{x^2+a^2}dx=displaystyle rac{pi}{2}e^{-ma}$
17.: $displaystyleint_{0}^{infty}displaystyle rac{sin mx}{x(x^2+a^2)}dx=displaystyle rac{pi}{2a^2}(1-e^{-ma})$
18.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bsin x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
19.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{a+bcos x}=displaystyle rac{2pi}{displaystyle sqrt{a^2-b^2}}$
20.: $displaystyleint_{0}^{pi/2}displaystyle rac{dx}{a+bcos x}=displaystyle rac{cos^{-1}(b/a)}{displaystyle sqrt{a^2-b^2}}$
21.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{(a+bsin x)^2}=int_{0}^{2... ...playstyle rac{dx}{(a+bcos x)^2}=displaystyle rac{2pi a}{(a^2-b^2)^{3/2}}$
22.: $displaystyleint_{0}^{2pi}displaystyle rac{dx}{1-2acos x+a^2}=displaystyle rac{2pi}{1-a^2},hspace{.2in}0<a<1$
23.: $displaystyleint_{0}^{pi}displaystyle rac{x sin x dx}{1-2acos x +a^2}=l... ...mid amid <1\ piln(1+1/a) &hspace{.3in} mid amid >1 end{array} ight.$
24.