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Differntial equation
En réponse à Souad Ayadi
Differential Equation نصائح لزملائي: علاقة المعدلات التفاضلية بالفزياء
par KAWTHER KLALDI,
1. Know their importance:
Differential equations are the language of change in physics – they describe motion, heat, electricity, waves, and more.
2. Focus on the basics:
· First-order equations (separation of variables).
· Linear second-order equations with constant coefficients.
· Learn the method of undetermined coefficients for non-homogeneous equations.
3. Connect math to physics:
· First derivative → velocity / current.
· Second derivative → acceleration / force.
· Exponential solution → growth or decay.
· Periodic solution → vibration or wave.
4. Steps to solve any problem:
1. Understand the physical system.
2. Write the appropriate mathematical equation.
3. Solve it.
4. Interpret the result physically.
5. Check the units.
5. Benefit from teamwork:
· Join study groups.
· Ask questions and share solutions.
· Explain concepts to classmates to deepen your understanding.
6. Practice on simple systems:
· Motion of an object with air resistance.
· Discharging a capacitor.
· Spring and mass system.
7. Prepare for advanced courses:
These equations are the foundation for analytical mechanics, quantum physics and wave theory.
Differential equations are the language of change in physics – they describe motion, heat, electricity, waves, and more.
2. Focus on the basics:
· First-order equations (separation of variables).
· Linear second-order equations with constant coefficients.
· Learn the method of undetermined coefficients for non-homogeneous equations.
3. Connect math to physics:
· First derivative → velocity / current.
· Second derivative → acceleration / force.
· Exponential solution → growth or decay.
· Periodic solution → vibration or wave.
4. Steps to solve any problem:
1. Understand the physical system.
2. Write the appropriate mathematical equation.
3. Solve it.
4. Interpret the result physically.
5. Check the units.
5. Benefit from teamwork:
· Join study groups.
· Ask questions and share solutions.
· Explain concepts to classmates to deepen your understanding.
6. Practice on simple systems:
· Motion of an object with air resistance.
· Discharging a capacitor.
· Spring and mass system.
7. Prepare for advanced courses:
These equations are the foundation for analytical mechanics, quantum physics and wave theory.
Differential equations are not just mathematics, they are a way to describe the universe.
Always try to see the mathematics behind the physical phenomenon this is the spirit of a successful physicist.
Always try to see the mathematics behind the physical phenomenon this is the spirit of a successful physicist.
How many type of differential equations that we have to know this year ?
This year we will study two types of differential equations:
1. Ordinary Differential Equations (ODEs)
2. Partial Differential Equations (PDEs)
1. Ordinary Differential Equations (ODEs)
2. Partial Differential Equations (PDEs)
In ODEs 2 methods
Bernoulli and exact equation ?
Bernoulli and exact equation ?
Not adjusting
Methods for solving Ordinary Differential Equations (ODEs):
1. For first-order equations:
· Separation of variables.
· Linear equations.
· Homogeneous equations.
· Bernoulli equation.
2. For second-order equations:
· Linear homogeneous equations with constant coefficients (using the characteristic equation).
· Linear non-homogeneous equations with constant coefficients.
هذا ما اعرفه و الله اعلم اذا كان لديك او لادا اي شخص منكم اجابه اخرى يفدنا و شكرا 🌹
Methods for solving Ordinary Differential Equations (ODEs):
1. For first-order equations:
· Separation of variables.
· Linear equations.
· Homogeneous equations.
· Bernoulli equation.
2. For second-order equations:
· Linear homogeneous equations with constant coefficients (using the characteristic equation).
· Linear non-homogeneous equations with constant coefficients.
هذا ما اعرفه و الله اعلم اذا كان لديك او لادا اي شخص منكم اجابه اخرى يفدنا و شكرا 🌹
Everything is clear
شكرا🥰
على لرحب
اردات ان أدعم اجباتي بصور كتبت فيهم الشرح بطريقة واضحة لكن مساحة الملفات المسموح بها صغيرة حوالي 500كيلو بايت
اردات ان أدعم اجباتي بصور كتبت فيهم الشرح بطريقة واضحة لكن مساحة الملفات المسموح بها صغيرة حوالي 500كيلو بايت
Do exact differential equations belong to PDEs or ODEs?
هل المعادلات التفاضلية التامة تنتمي الى PDEs او ODEs
هل المعادلات التفاضلية التامة تنتمي الى PDEs او ODEs
Exact differential equations belong to ODEs (Ordinary Differential Equations).
While you use partial derivatives to test for exactness and to find the potential function F(x,y), the equation itself describes a relationship between one independent variable (usually x) and one dependent variable
.
Key Summary
Classification: ODE.
Format: M(x, y)dx + N(x, y)dy = 0.
Why the confusion?: We use partial derivatives (\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}) as a tool to solve them, but the solution y(x) is a function of a single variable.
While you use partial derivatives to test for exactness and to find the potential function F(x,y), the equation itself describes a relationship between one independent variable (usually x) and one dependent variable
Key Summary
Classification: ODE.
Format: M(x, y)dx + N(x, y)dy = 0.
Why the confusion?: We use partial derivatives (\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}) as a tool to solve them, but the solution y(x) is a function of a single variable.
If we have verified the exactness condition, when searching for F , can we start by integrating N with respect to y instead of integrating M with respect to x ? And what are the steps in this case?
Yes, you can. Here is the concise process:
The Steps
Integrate N: Find F(x, y) = \int N \, dy + g(x).
Differentiate by x: Find \frac{\partial F}{\partial x} and set it equal to M.
Solve for g(x): Isolate g'(x) and integrate it with respect to x.
Final Answer: Plug g(x) back in and set F(x, y) = C.
The Steps
Integrate N: Find F(x, y) = \int N \, dy + g(x).
Differentiate by x: Find \frac{\partial F}{\partial x} and set it equal to M.
Solve for g(x): Isolate g'(x) and integrate it with respect to x.
Final Answer: Plug g(x) back in and set F(x, y) = C.