Solving separable differential equations
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Solving separable differential equations
- 1. 𝑑𝑦 𝑑𝑥 = 5𝑦 𝑑𝑦 𝑑𝑥 − 5𝑦 = 0 𝑒−5∫ 𝑑𝑥 = 𝑒−5𝑥 𝑦𝑐 = 𝐶𝑒5∫ 𝑑𝑥 = 𝐶𝑒5𝑥 𝑦𝑝 = 1 𝑒−5𝑥 ∫ 𝑒−5𝑥(0) 𝑑𝑥 = 0 𝑦 = 𝐶𝑒5𝑥 + 0 = 𝐶𝑒5𝑥 𝑑𝑦 𝑑𝑥 = 2𝑦 𝑑𝑦 𝑑𝑥 − 2𝑦 = 0 𝑒−2∫ 𝑑𝑥 = 𝑒−2𝑥 𝑦𝑐 = 𝐶𝑒2∫ 𝑑𝑥 = 𝐶𝑒2𝑥 𝑦𝑝 = 1 𝑒−2𝑥 ∫ 𝑒−2𝑥(0) 𝑑𝑥 = 0 𝑦 = 𝐶𝑒2𝑥 + 0 = 𝐶𝑒2𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 𝑒3𝑥 𝑒∫ 𝑑𝑥 = 𝑒 𝑥 𝑦𝑐 = 𝐶𝑒−∫ 𝑑𝑥 = 𝐶𝑒−𝑥 𝑦𝑝 = 1 𝑒 𝑥 ∫ 𝑒 𝑥 𝑒3𝑥 𝑑𝑥 = 1 𝑒 𝑥 ∫ 𝑒4𝑥 𝑑𝑥 = 𝑒3𝑥 4 𝑦 = 𝐶𝑒−𝑥 + 𝑒3𝑥 4 3 𝑑𝑦 𝑑𝑥 + 12𝑦 = 4 𝑑𝑦 𝑑𝑥 + 4𝑦 = 4 3 𝑒4 ∫ 𝑑𝑥 = 𝑒4𝑥 𝑦𝑐 = 𝐶𝑒−4∫ 𝑑𝑥 = 𝐶𝑒−4𝑥 𝑦𝑝 = 1 𝑒4𝑥 ∫ 𝑒4𝑥 ( 4 3 ) 𝑑𝑥 = 1 3 𝑦 = 𝐶𝑒−4𝑥 + 1 3 𝑦` + 3𝑥2 𝑦 = 𝑥2 𝑑𝑦 𝑑𝑥 + 3𝑥2 𝑦 = 𝑥2 𝑒3 ∫ 𝑥2 𝑑𝑥 = 𝑒 𝑥3 𝑦𝑐 = 𝐶𝑒−3∫ 𝑥2 𝑑𝑥 = 𝐶𝑒−𝑥3 𝑦 𝑝= 1 𝑒 𝑥3 ∫ 𝑒 𝑥3 𝑥2 𝑑𝑥= 1 3 𝑦 = 𝐶𝑒−𝑥3 + 1 3