y′ = 2x + 3y x′ = 7x − 4y x(0) = 2 y(0) = −1 sisteminin ¸c¨oz¨um¨un¨u bulunuz. (Lineer Denk. Sis.)



Answer to a math question y′ = 2x + 3y x′ = 7x − 4y x(0) = 2 y(0) = −1 sisteminin ¸c¨oz¨um¨un¨u bulunuz. (Lineer Denk. Sis.)

Expert avatar
74 Answers
نظرا لنظام المعادلات الخطية: ص′ = 2س + 3ص س′ = 7س – 4ص يمكننا استخدام طريقة المعادلات التفاضلية لإيجاد حل نظام المعادلات هذا. كخطوة أولى، دعونا نجد الصورة المتجانسة للمعادلات: ص' - 3ص = 2س س' - 7س = -4ص يمكن كتابة هذا النظام من المعادلات المتجانسة على شكل مصفوفة تمثل المعادلات المذكورة أعلاه على النحو التالي: [d/dt [x(t)] ] [ -7 4 ] [ x(t) ] [0] [d/dt [y(t)] ] = [ -2 3 ] [ y(t) ] + [0] معادلة المصفوفة هذه عبارة عن نظام من المعادلات الخطية المتجانسة من الدرجة الأولى التي تحتوي على المتجه [x(t), y(t)]. يمكن الحصول على حل معادلة المصفوفة هذه باستخدام القيم الذاتية والمتجهات الذاتية. لحساب القيم الذاتية، يتم حل المعادلة المميزة للمصفوفة: ديت (أ - μI) = 0 هنا A هي المصفوفة التي تحتوي على معاملات المصفوفة، و lect هي رمز القيم الذاتية، و I هي مصفوفة الهوية. المعادلة المميزة للمصفوفة هي: ديت ([-7-4 4] [-2 3-]) = 0 عندما نحل هذه المعادلة نجد قيمتين مختلفتين: ₁ = 1 ₂ = -3 لكل قيمة ذاتية، يمكننا حساب المتجهات الذاتية. لهذا، يتم حل المعادلة (A - αI) * v = 0، حيث v هو المتجه الذاتي. من أجل ς₁ = 1، (A - ς₁I) * v₁ = 0 [-8 4] [v₁₁] = [0] [-2 2] [v₁₂] = [0] عندما نحل هذه المعادلة، نحصل على المتجه الذاتي v₁ = [1، 2]. بالنسبة إلى ς₂ = -3، (A - ς₂I) * v₂ = 0 [4 4] [v₂₁] = [0] [-2 -6] [v₂₂] = [0] عندما نحل هذه المعادلة، نحصل على المتجه الذاتي v₂ = [-2، 1]. في الخطوة الأخيرة، نستخدم المتجهات الذاتية للحصول على الحل العام: [x(t)] [1 * e^χ₁t -2 * e₁₂t] [C₁] [y(t)] = [2 * e^lect₁t 1 * e^lect₂t] [C₂] هنا C₁ وC₂ ثوابت تمثل الظروف الأولية في الوقت t = 0. نظرًا لأن الشروط الأولية معطاة كـ x(0) = 2 و y(0) = -1، فيمكننا إيجاد قيم C₁ وC₂: [x(0)] [1 -2] [C₁] [2] [y(0)] = [2 1] [C₂] = [-1] عندما نحل هذه المعادلة، نحصل على C₁ = 0 وC₂ = -1. أخيرًا، باستبدال القيمتين C₁ وC₂ في الحل العام، نحصل على الحل: [x(t)] [1 * e^t -2 * e^(-3t)] [0] [e^t - 2 * e^(-3t)] [y(t)] = [2 * e^t 1 * e^(-3t)] [-1] = [2e^t - e^(-3t) - 1] وبهذه الطريقة، نحصل على حل نظام المعادلات الخطية المحدد.

Frequently asked questions (FAQs)
What is the inverse sine of 1?
What is the limit of (x^2 + 5)/(3x - 2) as x approaches 2?
What is the radian measure of a quarter turn?
New questions in Mathematics
If we have the sequence: 3, 6, 12, 24 Please determine the 14th term.
Solution to the equation y'' - y' - 6y = 0
Additionally, the boss asked Armando to determine how many toy sales branches he would have in the fifteenth year, knowing that the first year they started with two branches, by the second they already had 5 branches and, by the third year, they had 8 branches. From the above, determine the number of branches it will have for the fifteenth year.
You are planning to buy a car worth $20,000. Which of the two deals described below would you choose, both with a 48-month term? (NB: estimate the monthly payment of each offer). i) the dealer offers to take 10% off the price, then lend you the balance at an annual percentage rate (APR) of 9%, monthly compounding. ii) the dealer offers to lend you $20,000 (i.e., no discount) at an APR of 3%, monthly compounding.
How many different ways can a psychology student select 5 subjects from a pool of 20 subjects and assign each one to a different experiment?
A pair of die is thrown and the absolute difference of the two scores is recorded. What is the probability of the absolute difference being 4 or more?
7. Find the equation of the line passing through the points (−4,−2) 𝑎𝑛𝑑 (3,6), give the equation in the form 𝑎𝑥+𝑏𝑦+𝑐=0, where 𝑎,𝑏,𝑐 are whole numbers and 𝑎>0.
The market for economics textbooks is represented by the following supply and demand equations: P = 5 + 2Qs P = 20 - Qd Where P is the price in £s and Qs and Qd are the quantities supplied and demanded in thousands. What is the equilibrium price?
A machine produces 255 bolts in 24 minutes. At the same rate, how many bolts would be produced in 40 minutes?
The function h(t)=-5t^2+20t+60 models the height in meters of a ball t seconds after it’s thrown . Which describe the intercepts and vertex of this function
The probability of growing a seedling from a seed is 0.62. How many seeds do I need to plant so that the probability of growing at least one seedling is greater than or equal to 0.87?
For what values of m is point P (m, 1 - 2m) in the 2⁰ quadrant?
We have received our p&l statement back from accounts. The board has asked for an innovation hub. What items should we prioritise reviewing to decide if we can afford an innovation hub?
Calculate NPV, IRR and PAYBACK through a cash flow for a period of five years, with discount rate of: a) 10% b) 12% c) 15% initial annual cost $41,400,000
calculate the product of 4 and 1/8
Carmen's age was twice as old as Luis was when Carmen was Luis's age. When Luis is Carmen's age, their ages will add up to 112.
I have a complex function I would like to integrate over. I can use two approaches and they should give the same solution. If I want to find the contour integral ∫𝛾𝑧¯𝑑𝑧 for where 𝛾 is the circle |𝑧−𝑖|=3 oriented counterclockwise I get the following: ∫2𝜋0𝑖+3𝑒𝑖𝑡⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯𝑑(𝑖+3𝑒𝑖𝑡)=∫2𝜋03𝑖(−𝑖+3𝑒−𝑖𝑡)𝑒𝑖𝑡𝑑𝑡=18𝜋𝑖 If I directly apply the Residue Theorem, I would get ∫𝛾𝑧¯𝑑𝑧=2𝜋𝑖Res(𝑓,𝑧=0)=2𝜋𝑖
5 1/9 + 2 2/3