MATH 202 Final SAMPLE, Spring 2019

Sample Final Exam By Tarek Hajj Shehadi Problem I : Consider the following ODE : 𝑑𝑦 𝑑𝑥 = 𝑎(𝑥)𝑦 + 𝑏(𝑥)𝑦2 + 𝑐(𝑥) a) Assume 𝑎 = 1, 𝑏 = 2, 𝑐 = −3 find a solution to the ODE b) Assume 𝑎 = 0 and that 𝑐(𝑥) = 𝑥𝑛 solve the ODE for 𝑛 = −2 [Hint : use the substitution 𝑦 = 1 𝑧 ] c) If a particular solution 𝑦1 is known , the general solution is given by 𝑦 = 𝑦1 + 𝑢 show that 𝑑𝑢 𝑑𝑥 = 𝑏(𝑥)𝑢2 + [2𝑏(𝑥)𝑦1 + 𝑎(𝑥)]𝑢 Problem II : Solve the following ODEs : a) 𝑥′𝑡(𝑥′ + 2) = 𝑥 b) 𝑦(2𝑥4 + 𝑦) 𝑑𝑦 𝑑𝑥 = (1 − 4𝑥𝑦2)𝑥2 c) 𝑦” + 𝑦 = 8 cos(x) cos(2x) by : i) knowing that cos(a)cos(b) = cos(𝑎−𝑏)+cos(𝑎+𝑏) 2 and using variation of parameters ii) using the Wronskian find the general solution d) 𝑦” − 2𝑦′ + 2𝑦 = 𝑥 + 1 y(0) = a ; y’(0) = 0 Problem III : The following ODE is not exact . DERIVE an integrating factor to make it exact : (2𝑥 + 1)(cos(y))dx + (x2 + 𝑥)(sin(𝑦)) + tan(𝑦) 𝑑𝑦 = 0 Problem IV : Use 𝑥 = 𝑒𝑡 to convert the Cauchy-Euler equation into a differential equation with constant coefficients then solve the ODE : 𝑥2𝑦" − 𝑥𝑦′ + 𝑦 = ln(𝑥) Problem V : i) Using the method of Frobenius solve the ODE : 3𝑥𝑦" + 𝑦′ − 𝑦 = 0 about 𝑥 = 0 ii) Using the taylor series of cos(x) solve the ODE : 𝑦" + (cos (𝑥))𝑦 = 0 Problem VI : Using Laplace Transforms solve the following ODEs : a) 𝑦” + 3𝑦′ + 2𝑦 = 𝑔(𝑡) ; 𝑦(0) = 0 𝑎𝑛𝑑 𝑦′(0) = 1 𝑔(𝑡) = {1 𝑖𝑓 0 ≤ 𝑡 < 1 0 𝑖𝑓 𝑡 ≥ 1 b) 𝑦′(𝑡) = 1 − ∫ 𝑦(𝑡 − 𝑣)𝑒−2𝑣𝑑𝑣 𝑡 0 c) 𝑦” + 2𝑦′ − 3𝑦 = 6𝑡2𝑒−3𝑡 [express in convolutions] Problem VII : Using eigenvalues solve the ODEs : a) 𝑋′ = ( 3 6 −1 −2) 𝑋 b) 𝑋′ = (3 −18 2 −9 ) 𝑋 c) 𝑋′ = ( 2 8 −1 −2) 𝑋 Problem VIII : Think of a way to solve this first order ODE : 𝑑𝑦 𝑑𝑥 = 1 𝑥(𝑥 − 𝑦)