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MASC 20001 Advanced Mathematics Exercises

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Advanced Mathematics (MASC 20001.2) – Spring – 2020 – CW (Assignment-1) – All – QP 1. Find the equation of the curve passing through the point (1, 1) whose differential equation is (𝑦 − 𝑦𝑥)𝑑𝑥 + (𝑥 + 𝑥𝑦)𝑑𝑦 = 0 (𝑦 − 𝑦𝑥)𝑑𝑥 + (𝑥 + 𝑥𝑦)𝑑𝑦 = 0 → 𝑦(1 − 𝑥)𝑑𝑥 + 𝑥(1 + 𝑦)𝑑𝑦 = 0 → 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑦 𝑥𝑦 → 𝑦(1 − 𝑥) 𝑥(1 + 𝑦) 1−𝑥 1+𝑦 𝑑𝑥 + 𝑑𝑦 = 0 → 𝑑𝑥 + 𝑑𝑦 = 0 → 𝑥𝑦 𝑥𝑦 𝑥 𝑦 1 1 1 1 ( − 1) 𝑑𝑥 + ( + 1) 𝑑𝑦 = 0 → 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒 → ∫ ( − 1) 𝑑𝑥 + ∫ ( + 𝑦) 𝑑𝑦 = 𝑐1 → 𝑥 𝑦 𝑥 𝑦 ln|𝑥| − 𝑥 + 𝑐𝑥 + ln|𝑦| + 𝑦 + 𝑐𝑦 = 𝑐1 → ln|𝑥. 𝑦| + (𝑦 − 𝑥) = 𝑐 𝑁𝑜𝑤 𝑤𝑒 𝑝𝑙𝑢𝑔 𝑖𝑛 𝑥 = 1 𝑎𝑛𝑑 𝑦 = 1 → ln|1.1| + (1 − 1) = 𝑐 → 0 + 0 = 𝑐 → 𝑐 = 0 ∴ 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡ℎ𝑟𝑜𝑢𝑔ℎ (1, 1): ln|𝑥𝑦| + (𝑦 − 𝑥) = 0 𝐶𝑜𝑛𝑡𝑖𝑛𝑢𝑒 𝑡𝑜 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦: ln|𝑥𝑦| = (𝑥 − 𝑦) → 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑒 → 𝑒 ln|𝑥𝑦| = 𝑒 𝑦−𝑥 → 𝑥𝑦 = 𝑒 𝑥−𝑦 2. Solve the following differential equation by using suitable method: 𝑥(𝑥 − 1) 𝑑𝑦 − 𝑦 = 𝑥 2 (𝑥 − 1)2 ...
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