The document discusses ordinary differential equations (ODEs). It begins by introducing ODEs as important connections between pure and applied mathematics. It notes that all areas of engineering, physics, chemistry, and dynamics involve ODEs. It then defines an ODE as a relationship between a dependent variable and its derivatives with respect to an independent variable. Basic forms of ODEs and common types classified by order, linearity, homogeneity, and integrability are also outlined. Finally, it states that the lecture will discuss definitions of ODEs, basic methods for solving linear and quadratic equations, and some advanced solution techniques.

1. ‫التفاضلية‬ ‫المعادالت‬
‫العادية‬
Ordinary
Differential
Equations

2. ‫مقدمة‬
:
‫التفاضلية‬ ‫المعادالت‬ ‫تعتبر‬
‫العلوم‬ ‫بين‬ ‫الرابط‬ ‫وهي‬ ‫والتطبيقية‬ ‫البحتة‬ ‫الرياضيات‬ ‫في‬ ‫المهمة‬ ‫المواضيع‬ ‫من‬
‫والهندسية‬ ‫الرياضية‬
.
‫التفاضلية‬ ‫المعادالت‬ ‫أنواع‬ ‫من‬ ‫واالنشائية‬ ‫والميكانيكية‬ ‫الكهربائية‬ ‫الهندسة‬ ‫مواضيع‬ ‫تخلو‬ ‫فال‬
.
‫التفاعالت‬ ،‫الهندسية‬ ‫الحركة‬ ،‫السكاني‬ ‫النمو‬ ‫مثل‬ ،‫والظروف‬ ‫الظواهر‬ ‫من‬ ‫متنوعة‬ ‫مجموعة‬ ‫وحل‬ ‫لنمذجة‬ ‫قوية‬ ‫أداة‬ ‫تعتبر‬
‫الديناميكية‬ ‫واألنظمة‬ ،‫الكيميائية‬
.
‫التحليلية‬ ‫الحلول‬ ‫ذلك‬ ‫في‬ ‫بما‬ ،‫المعادالت‬ ‫هذه‬ ‫حلول‬ ‫لحساب‬ ‫مختلفة‬ ‫طرق‬ ‫توجد‬
‫والحسابية‬
‫أويلر‬ ‫طريقة‬ ‫مثل‬ ‫متقدمة‬ ‫تقنيات‬ ‫باستخدام‬
.
‫المعادالت‬ ‫لحل‬ ‫المختلفة‬ ‫الطرق‬ ‫وماهي‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تعريف‬ ‫سنناقش‬ ، ‫المحاظرة‬ ‫هذا‬ ‫في‬
.
‫والتربيعية‬ ‫الخطية‬ ‫المعادالت‬ ‫لحل‬ ‫األساسية‬ ‫الطرق‬ ‫سنناقش‬ ‫ثم‬
.
‫الم‬ ‫لحل‬ ‫المتقدمة‬ ‫الطرق‬ ‫بعض‬ ‫سنناقش‬ ،‫ا‬ً‫أخير‬
‫غير‬ ‫عادالت‬
‫الخطية‬
.

3. ‫التفاضلي‬ ‫المعادلة‬ ‫تعريف‬
‫ة‬
:
‫وليكن‬ ‫مستقل‬ ‫متغير‬ ‫بين‬ ‫تساوي‬ ‫عالقة‬ ‫هي‬ ‫التفاضلية‬ ‫المعادلة‬
x
‫وليكن‬ ‫تابع‬ ‫ومتغير‬
y
‫التابع‬ ‫المتغير‬ ‫لهذا‬ ‫التفاضلية‬ ‫المشتقات‬ ‫من‬ ‫اكثر‬ ‫او‬ ‫واحد‬
.
‫ال‬
‫لـ‬ ‫العامة‬ ‫صورة‬
ODE
:
F(y,y′,y′′,…,y(n))=0
‫التفاضل‬ ‫الدورات‬ ‫بين‬ ‫رياضي‬ ‫توازن‬ ‫شكل‬ ‫على‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تأتي‬
‫معينة‬ ‫لدوال‬ ‫ية‬
‫المسافة‬ ‫أو‬ ‫الزمن‬ ‫مثل‬ ‫متغيرات‬ ‫على‬ ‫تحتوي‬ ‫وقد‬
.
‫ب‬ ‫المعادالت‬ ‫هذه‬ ‫تكون‬ ‫أن‬ ‫يمكن‬
‫سيطة‬
‫تصفها‬ ‫التي‬ ‫الظاهرة‬ ‫نوع‬ ‫على‬ ‫تعتمدا‬ ‫معقدة‬ ‫أو‬
.

4. ‫التفاضلية‬ ‫المعادالت‬ ‫أنواع‬
:
‫منها‬ ‫معاير‬ ‫عدة‬ ‫حسب‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تصنيف‬ ‫يمكن‬
:

‫الرتبة‬
:
‫فيها‬ ‫موجودة‬ ‫مشتقة‬ ‫أعلى‬ ‫بأنها‬ ‫التفاضلية‬ ‫المعادلة‬ ‫مرتبة‬ ‫عرف‬ُ‫ت‬
.

‫النوع‬
:
‫خطية‬ ‫غير‬ ‫أو‬ ‫خطية‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تكون‬ ‫أن‬ ‫يمكن‬
.

‫الحالة‬
:
‫متجانسة‬ ‫غير‬ ‫أو‬ ‫متجانسة‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تكون‬ ‫أن‬ ‫يمكن‬

‫التكامل‬
:
‫للتكامل‬ ‫قابلة‬ ‫غير‬ ‫أو‬ ‫للتكامل‬ ‫قابلة‬ ‫التفاضلية‬ ‫المعادالت‬ ‫تكون‬ ‫أن‬ ‫يمكن‬
.

6. Preparing for
the exam
02

7. Study planning (I)
01
Date of the exam
(note it in your planner)
Find out
about:
02
Type of exam
(oral, practical, multiple-choice...)
03
Information
Content that may be included in the exam

8. Study planning (II)
1. Count the days you have to study from the day you are notified until the day of the
exam
2. Take into account other tasks, homework or activities you have during those days
and calculate the time (hours per day) you will have for studying
3. Break down the information by topics or by sections
4. Use a weekly planner and write in it which sections you are going to study each day
from the day you start until the day of the exam
5. Organize your study, making sure that the last few days are for review

9. During the
exam
03

10. Time planning
in the exam
Take a few minutes to get organized:
1. Find out how much time you have to
take the exam
2. Keep a watch or clock handy to keep
track of the time
3. Take a first look at the exam: how
many questions there are, what kind,
which ones you know and which
ones you don't know
4. Determine how much time to devote
to each question
5. To avoid possible mistakes, solve
your doubts before you start by
asking the teacher the questions you
need to ask

11. Start to answer:
Read each question
carefully
01
Start by answering the
ones you know best first
02
Continue with those you
partially know
03
Review your answers to
make sure you haven't
forgotten anything
04
Save the questions you
don't know for the end
05
Do a final review before
turning it in
06

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