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
126 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)
Math question: What is the equation of a basic shape of a graph that resembles a straight line that neither increases nor decreases?
+
What is the limit as x approaches 3 of [(2x^2 + x - 1)/(x - 3)]?
+
Question: What is the domain and range of the logarithmic function 𝑓(𝑥) = log(𝑥), where 𝑥 > 0?
+
New questions in Mathematics
1/2x +3 <4x-7
If O(3,-2) is reflected across x = 2. What are the coordinates of O
X^2 = 25
Determine the equations of the recipes that pass through the following pairs of points P1 (2;-1) and p2 (4;-1)
[(36,000,000)(0.000003)^2]divided(0.00000006)
The data set (75, 85, 58, 72, 70, 75) is a random sample from the normal distribution No(µ, σ). Determine a 95% two-sided confidence interval for the mean µ .
prove that if n odd integer then n^2+5 is even
-3(-4x+5)=-6(7x-8)+9-10x
. What will be the osmotic pressure of a solution that was prepared at 91°F by dissolving 534 grams of aluminum hydroxide in enough water to generate 2.784 ml of solution.
Quadratic equation 2X = 15/X + 7
5x+13+7x-10=99
(X+2)(x+3)=4x+18
Determine the Linear function whose graph passes through the points (6, -2) and has slope 3.
A buyer purchased a North Carolina home for $475,250. The seller allowed the buyer to assume his first small mortgage with a loan balance of $110,000. How much is the excise tax paid in the transaction? $951 $729.50 $950.50 $221 none of the above
a) Statistics scores are normally distributed with the mean of 75 and standard deviation of 7. What is the probability that a student scores between 80 and 88
The average undergraduate cost per tuition, fees, room, and board for all institutions last year was $26,025. A random sample of 40 institutions of higher learning this year indicated that the mean tuition, fees, room, and board for the sample was $27,690, and the population standard deviation is $5492. At the 0.05 level of significance, is there sufficient evidence that the cost has increased? (Remember to follow the steps in hypothesis testing)
8/9 divided by 10/6
How many moles are there in 235 grams of potassium thiosulfate pentahydrate? K2S2O3*5(H2O)
A person travels by car from one city to another with different constant speeds between pairs of cities. She drives for 55.0 min at 100.0 km/h, 14.0 min at 65.0 km/h, and 45.0 min at 60.0 km/h and spends 20.0 min eating lunch and buying gas. (a) Determine the average speed for the trip.
f(r) = 1/r+9 find f(x^2) + 1