Question

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

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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.)

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Hermann
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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.

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