Ahmed Al Ansaary

H.O.L.D.Es.W.C.Cs

ahansaary — Sun, 17 Mar 2013 16:11:37 GMT

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2nd order DE's reducible to 1st order DE's

ahansaary — Fri, 15 Mar 2013 19:23:04 GMT

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ODE of 1st order and 1st degree F(x,y,y')

ahansaary — Wed, 13 Mar 2013 22:11:06 GMT

الصورة العامة - y' = F(x,y)

or

M(x,y)dx + N(x,y)dy = 0

1- Separable equations :

ويتم فيها فصل x عن y (فصل المتغيرات)

to solve the separable equation :

1. set y' = dy/dx

dy/dx = f(x)g(y)

2. dy/g(y) = f(x)dx

3. by integration we obtain the solution .

:: Special case ::

- If the DE in the form y' = F ( ax + by + c )

to solve this equation we set [ ax + by + c = z ]

2- Homogeneous equations :

F(x,y) is called homogeneous of degree if F( tx, ty ) = tⁿ F(x,y)

إذا كان كل حد من حدود المعادلة من نفس الدرجة تبقى homogeneous

or x/y or y/x

a DE     M(x,y)dx + N(x,y)dy = 0

is called homogeneous if M(x,y) & N(x,y) are homogeneous of the same degree

To solve the homogeneous DE :-

1. set y = vx

2. y' = v + xv'

3. بالتعويض هاتتحول المعادلة الى separable

:: Differential Equations reducible either separable or homogeneous ::

if the differential equations in the form

a₁x + b₁y + c₁
y' = ـــــــــــــــــــــــــــــــــــــــ
a₂x + b₂y + c₂

1- parallel :

if a₁/b₁ = a₂/b₂

we set
a₁x + b₁y = z
or
a₂x + b₂y = z

then the equation is reduced to separable .

2- intersection :

if a₁/b₁ != a₂/b₂

we solve the two equation to find the point of intersection (h,k)

then

we set     x = t + h , y = s + k

and       dy/dx = ds/dt

by substitution , the equation is reduced to homogeneous .

:: Exact differential equation ::

A DE M(x,y)dx + N(x,y)dy = 0 لازم موجب

if dM/dy = dN/dx the equation is called exact .

the solution of the exact DE is in the form

حيث x_o , y_o قيم ابتدائية اختيارية يتم اختيارها على اساس جعل المعادلة أسهل ما يمكن ولا تعتمد على بعضها وعادة ما نختارها بـ 0 أو 1 في حالة ln

:: Integrating Factor عامل المكاملة ::

if dM/dy != dN/dx the DE is not exact .

- we can transform this equation to exact by multiplying it by integrating factor M ميو

- to find the integrating factor - two cases :-

:: Linear differential equations ::

:: Bernoulli 's equation ::

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Introduction

ahansaary — Wed, 13 Mar 2013 21:22:14 GMT

Differential Equations

divided two branches :

1- ordinary differential equations .
which containing only independent variable .

2- partial differential equations .
contains more than one independent variable .

ordinary differential equations ( ODEs ) :
an equation involves an unknown function of one variable and some its derivatives .

:: Basic concepts ::

1- order of DE . is the highest order derivative .

2- degree of DE . is the highest power of the highest order derivative . هو أعلى أس لأعلى رتبة تفاضل

:: Formation of differential equation ::

the DE can be formed by
التفاضل وحذف الثوابت الإختيارية
رتبة المعادلة يساوي عدد الثوابت

:: solution of ODEs ::

حل المعادلات التفاضلية : هو أي دالة تحقق المعادلة التفاضلية

sol of ODE :

1- general solution .
2- particular solution .

الحل العام : هو الحل الذي يحتوي على عدد من الثوابت الإختيارية يساوي رتبة المعادلة التفاضلية .

الحل الخاص : هو حل خاص من الحل العام عندما نوجد الثوابت الإختيارية في الحل العام .

:: Initial value problem ::

any DE which contain initial conditions .
دائما وأبدا حل ivp يكون بالحل الخاص

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