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)
Find the sum of the major and minor axis lengths of an ellipse with equation (x/3)^2 + (y/4)^2 = 1.
+
Math Question: What is the limit of [(3x^2 - 2x + 5)/(x^2 - 3x + 2)] as x approaches 2?
+
Question: How many ways can a committee of 4 be formed from a group of 10 candidates?
+
New questions in Mathematics
Given the vectors: a = (2m – 3n, 4n – m) and b = (2, -3), find the values of m and n that make: a = 5 b.
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.
There are 162 students enrolled in the basic mathematics course. If the number of women is 8 times the number of men, how many women are there in the basic mathematics course?
Find the derivatives for y=X+1/X-1
A study reports the following final notation: F (3, 32) = 9.50, p < .05. How many total participants were involved in this study? Group of answer choices 34 32 36
determine the polynomial F of degree 2 that interpolates. f at points (0;1) (2;5) (4;6). calculate F(0.8). Note: Using the polynomial expression with difference operator.
The physician orders 15mg of tramadol(liquid). On hand is 30mg/2mL vials. How many mL will the MA administer?
How many square feet of floor area are there in three two-storey apartment houses, each of which is 38 feet wide and 76 feet long?
DuocUC 2) The cost C, in pesos, for the production of x meters of a certain fabric can be calculated through the function: (x+185) C(x)=81300-6x+ 20000 a) It is known that C(90) 5.344. Interpret this result. (2 points) b) Calculate C'(x) (2 points) 3 x²+111x-0.87 20000 2000 c) Function C calculates the cost while producing a maximum of 500 meters of fabric. Determine the values of x at which the cost of production is increasing and the values of x at which the cost is decreasing. (3 points) d) If a maximum of 500 meters of fabric are produced, what is the minimum production cost? (
From 1975 through 2020 the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 34 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 800? Assume the standard deviation is 1539
The two sides of the triangle are 12 cm and 5 cm, and the angle between the sides is 60°. Cover the area of ​​the triangle!
The sick-leave time of employees in a firm in a month is normally with a mean of 100 hours and a standard deviation of 20 hours. Find the probability that the sick-leave time of an employee in a month exceeds 130 hours.
-1%2F2x-4%3D18
A Smooth Plane is listed for $195.00. Discounts of 12% and 10% are allowed. If the customer pays cash within 30 days, an additional discount of 3% is granted. What is the cost if a carpenter takes advantage of all the discounts offered?
2.380× (1+0.05) / 0.95−0.05
1. A jeweler has two gold bars, with 80% purity and the other with 95% purity. How much of each must be melted to obtain a 5 kilo ingot with 86% purity?
For the numbers below, select a number at random and find the probability that: a. The number is even b. The sum of the number’s digit is even c. The number is greater than 50 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
Salut👋🏻 Appuie sur "Créer une nouvelle tâche" pour envoyer ton problème de mathématiques. Un de nos experts commencera à travailler dessus immédiatement !
16-(x²+x+2)²
g(x)=3(x+8). What is the value of g(12)