Math 210 Differential Equations Mock Final Dec *************************************************************** 1. Initial Value Problems

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1 Math 210 Differential Equations Mock Final Dec *************************************************************** 1. Initial Value Problems 1. Construct the explicit solution for the following initial value problem: dx = x 5 y 2,y(0) =3. 2. Construct the explicit solution for the following initial value problem: 2 dx + y = ex,y(0) =2. 3. Solve the following initial value problem explicitly. dp dt 4. Solve the initial value problem = p 2p 2,p(0) =10 ty + 2y = 4t 2, y(1) =2. 5. Find the explicit solution for the initial value problem 6. Solve the given initial value problem. 7. Solve the given initial value problem. dx = 3x2 + 4x + 2,y(0) = 1 2(y 1) y 2y = 0;y(0) =0,y (0) =1 y y = 2, y(0) =1, y (0) =2. 8. Find the solution of the initial value problem 16y 8y + 145y = 0, y(0) = 2, y (0) =1. 2. General Solutions 1.Construct the general solution (explicit) for the following differential equation: (e x y + 1)dx +(e x 1) = 0 2.Solve the following differential equation explicitly.

2 dx = x + y 1 (Hint: use a suitable substitution.) 3.Solve the following differential equation explicitly. dx + y x = x2 y 2 4. Find an integrating factor and solve the differential equation. 5.Find the general solution to 6. Find the general solution of ydx +(2x ye y ) = 0 y 2y + 5y = 0 y (4) y = Word Problems 1.A swimming pool whose volume is 20,000 gal contains water that is 0.01% chlorine. Starting at t = 0, city water containing 0.001% chlorine is pumped into the pool at a rate of 10 gal/min. The pool water flows out at the same rate. What is the percentage of chlorine in the pool after 1 hr? When will the pool water be 0.002% chlorine? 2.An object of mass 1kg is given an initial upward velocity 30m/s from the ground level and allowed to rise vertically under the influence of gravity. Assuming the gravitational force is constant and the force due to air resistance is proportional to the velocity of the object with k = 2kg/sec, determine the time when the object reaches the peak. 3. An object of mass 1kg is given an initial upward velocity 30m/s from the ground level and allowed to rise vertically under the influence of gravity. Assuming the gravitational force is constant and the force due to air resistance is proportional to the velocity of the object with k = 2kg/sec, determine the maximum height that the object is able to reach. You can leave the answer in the form of an explicit integral. 4. A tank initially contains 120 liters of pure water. A mixture containing a concentration of γ g/liter of salt enters the tank at a rate of 2 liters/min, and the well-stirred mixture leaves the tank at the same rate. Find an expression in terms of γ for the amount of salt in the tank at any time t. Also find the limiting amount of salt in the tank as t. 5. A ball with mass 0.15 kg is thrown upward with initial velocity 20m/sec from the roof of a building 30 m high. Assume that there is a force due to air resistance of v /30, where the velocity v is measured in m/sec. (a) Find the maximum height above the ground that the ball reaches. (b) Find the time that the ball hits the ground. 4. Stability Analysis

3 Classify each equilibrium point as asymptotic stable, unstable, or semistable. 1. /dt = y 2 (4 y 2 ), < y 0 < 2. /dt = ay b y,a > 0,b > 0, y Reduction of Order 1. Given that f(x) =e 2x is a solution to y 4y + 4y = 0, use the method of reduction of order to determine a second linearly independent solution. 2.Find a second independent solution of the given equation t 2 y + 3ty + y = 0, t > 0; y 1 (t) =t 1 of 6. Wronskians 1. Show that if p is differentiable and p(t) >0, then the Wronskian W(t) of two solutions is W(t) =c/p(t), wherec is a constant. [p(t)y ] + q(t)y = 0 7. Method of Undetermined Coefficients 1. Find a particular solution of 2. Find a particular solution of y + 4y = 3cos2t. y + 9y = t 2 e t Variation of Parameters 1. Find a particular solution of y + y = tant, 0 < t <π/2 2. (a) Find a pair of fundamental solutions to t 2 y 2y = 0,t > 0 (Hint: Try y(t) =t α for some real number α.) (b) Using (a) and the method of variation of parameters to find a particular solution to t 2 y 2y = 3t 2 1,t > 0

4 9. Laplace Transform 1. Find the solution of the differential equation where Assume that the initial conditions are 2y + y + 2y = g(t), 1, 5 t<20, g(t) =u 5 (t) u 20 (t) = 0, 0 t<5 andt 20. y(0) =0,y (0) =0. 2. Find the solution of the given initial value problem using Laplace transform. y + 4y =δ(t 4π);y(0) =1/2,y (0) =0. 3. Find the solution of the initial value problem y + 4y = g(t), y(0) =3, y (0) = 1. (Hints: Convolution is required here) 4. Use the method of Laplace transform to find the solution of the differential equation satisfying the initial conditions y + y = 3sin2t y(0) =y (0) =0. 5. Use the method of Laplace transform to find the solution of the given initial value problem Here δ(t) is the Dirac delta function. y + y =δ(t 3)e t,y(0) =y (0) =0 10. Conversion into a Matrix Differential Equation of First Order 1. Convert the differential equation into a matrix differential equation of first order y (4) 2y (3) + 5y y = 0 2. Convert the differential system into a matrix differential equation of first order y + y 2 0 x + 3y + 4x = 0

5 Here x and y are the unknown functions. 11. Solving a Matrix Differential Equation and Fundamental Matrix 1. Find the general solution of 2. Solve the given initial value problem x x,x(0) = Given the system x, find the fundamental matrix Φ such that Φ(0) =I.