First and Second order DE, Exercises of Calculus for Engineers

Download First and Second order DE and more Exercises Calculus for Engineers in PDF only on Docsity! Problem Set 1 First order DE (Do at least 5 problems of each type) 1) 𝑦′ = 𝑦 − 𝑥, 𝑦(0) = 1/3 2) 𝑦′ = 𝑦4/5, 𝑦(0) = 1 3) 𝑦′ = 𝑥𝑦 𝑥2+4 4) 3𝑥𝑦′ − 1 = ln 𝑥 + 𝑦, 𝑦(1) = 2 5) 𝑦′ = 1−2𝑦−4𝑥 1+𝑦+2𝑥 by using 𝑦 + 2𝑥 = 𝑣 6) 𝑥3𝑦′ + 3𝑥2𝑦 = 1/𝑥 7) 𝑦′ = 𝑦 + 𝑦2, 𝑦(0) = 0 8) 𝑦′ = 2(𝑦 − 1) tanh 2𝑥 , 𝑦(0) = 4 9) Show that 𝑦 = 𝑐𝑒𝑥 + 𝑥 + 1 is the solution to 𝑦′ = 𝑦 − 𝑥 when 𝑦(0) = 3 10) 𝑦′𝑥 𝑙𝑛 𝑥 = 𝑦, 𝑦(3) = ln 81 11) 2𝑦𝑦′ + 𝑦2 sin 𝑥 = sin 𝑥, 𝑦(0) = √2 12) 𝑦′ + 2𝑦 = 𝑡2 − 𝑡 + 1, 𝑦(1) = 1 2 13) 𝑦′ = 𝑥𝑦3 √1+𝑥2 14) 𝑑𝑟 𝑑𝜃 = −𝑟 tan 𝜃 15) 𝑑𝑦 𝑑𝑥 = 𝑥(𝑒𝑥2 +2) 6𝑦2 ; 𝑦(0) = 1 16) 𝑑𝑦 𝑑𝑥 = 2𝑥(𝑦 − 5) 17) 𝑑𝑦 𝑑𝑥 = (1 + 𝑒−𝑥)(𝑦2 − 1) 18) sin 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 cos 𝑥 = sin 𝑥 19) 𝑥𝑦′ + 𝑦 = 𝑥2 + 1 20) 𝑑𝑦 𝑑𝑥 + 2𝑦 = cos 𝑥 21) 𝑥 𝑑𝑦 𝑑𝑥 − 4𝑦 = 𝑥6𝑒𝑥 22) 𝑑𝑦 𝑑𝑥 = 2√𝑦, ; 𝑦(1) = 0 23) 𝑑𝑦 𝑑𝑥 = 𝑥+1 8+2𝜋 sin(𝜋𝑦) 24) 𝑑𝑦 𝑑𝑥 = − 𝑥 𝑦−3 ; 𝑦(0) = 1 25) 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 1 𝑥 26) 𝑦′ + 2𝑥𝑦 = 1 27) 𝑑𝑦 𝑑𝑥 + cot 𝑥 𝑦 = cos 𝑥 28) 2(𝑦 − 4𝑥2) + 𝑥 𝑑𝑦 𝑑𝑥 = 0