Question

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

139

likes
697 views

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
Hermann
4.6
125 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 probability of flipping a fair coin 5 times and getting exactly 3 heads?
+
Question: Convert 42 feet to meters.
+
What is the equation of a line that passes through point (2, 5) with a slope of 3?
+
New questions in Mathematics
How to find the value of x and y which satisfy both equations x-2y=24 and 8x-y=117
-6n+5=-13
Since one of the three integers whose product is (-60) is (+4), write the values that two integers can take.
2x-y=5 x-y=4
Kayla has $8,836.00 in her savings account. The bank gives Kayla 5%of the amount of money in account as a customer bonus. What amount of money does the bank give Kayla? Justify your answer on a 6th grade level.
Identify a pattern in the list of numbers.Then use this pattern to find the next number. 37,31,25,19,13
Pedro had 80% of the amount needed to buy a game. Of this amount, you spent 15% on a watch and therefore, you will need to add another R$640.00 to purchase this game. Is the value of the game?
In a grocery store, when you take out 3 peppers and 4 carrots, there are 26 peppers and 46 carrots left. How many peppers and carrots were there initially?
Determine the general equation of the straight line that passes through the point P (2;-3) and is parallel to the straight line with the equation 5x – 2y 1 = 0:
A person decides to invest money in fixed income securities to redeem it at the end of 3 years. In this way, you make monthly deposits of R$300.00 in the 1st year, R$400.00 in the 2nd year and R$500.00 in the 3rd year. Calculate the amount, knowing that compound interest is 0.6% per month for the entire period. The answer is 15,828.60
TEST 123123+1236ttttt
Buffalo Company makes and sells shampoo. Each unit requires $1.40 labor costs, material costs per unit are $0.90 and other variable costs are $0.30. It sells shampoo for $4.45 to retailers. Fixed costs are $15,000. It sold 25,000 units in the current month. What is the Break-Even point in units? What is the Break-Even point in dollars? What is the contribution margin of Buffalo Company?
Express the trigonometric form of the complex z = -1 + i.
In measuring the internal radius of a circular sewer the measurement is 2% too large. If this measurement is then used to calculate the circular cross-sectional area of the pipe: Determine, by using the binomial theory, the percentage error that will occur compared to the true area.
5x+13+7x-10=99
A building lot is in the shape of a triangle with a base of 133 feet and a height of 76 feet. What is it's area in square feet?
simplify w+[6+(-5)]
16-(x²+x+2)²
Define excel and why we use it?
A gas is leaking at 3.5ft3/min in a room of 2.9m by 6.9ft by 15.7m. How long would it take (in seconds) for 22% of the room to reach the LFL, if the gas has a LFL of 2.51%?