MTH5123: Differential Equations Main Examination period 2019
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Main Examination period 2019
MTH5123: Differential Equations
Note for the rest of this exam paper ODE refers to ordinary differential equation.
Question 1. [20 marks] The equation of motion for a falling object of mass m is given by
dv
where g = 9.8 m/s2 is the acceleration due to gravity, γ is a constant called the drag coefficient and v = v(t) denotes the velocity of the object at time t. Assume
(a) Find the general solution to this differential equation with the given constants.
(b) Now find the specific solution satisfying the initial condition v(0) = 49.
(c) Draw integral curves in the t-v plane for various initial conditions, including
the initial condition v(0) = 49.
(d) Interpret your graph for this model, explaining briefly the behaviour of your
solutions as t 二→ 心 for different initial conditions.
Question 2. [20 marks] Consider the following first-order, linear, inhomogeneous initial value problem
二π/2 ╱ x ╱ π/2 .
(a) Find a solution y = y(x) to the initial value problem.
(b) Use the Picard-Lindelf Theorem to justify existence and uniqueness of solutions to the above IVP in an appropriate rectangular domain.
Question 3. [20 marks] Consider the differential equation given by
x2 + + ln |xy| = 0.
(a) Find all functions f (y) such that the differential equation becomes exact.
(b) For the function f which makes the differential equation exact and which further satisfies f (1) = 1, solve the equation in implicit form.
Question 4. [20 marks] Consider the following Euler-type equation
x2 二 2y = 0.
(a) Using x = et and z(t) = y(et), verify that (*) can be rewritten as 二 z˙ 二 2z = 0.
Use this equation to find the general solution y = y(x) to (*).
(b) Next, consider the Boundary Value Problem (BVP) for the second order inhomogeneous differential equation
(*)
x2 二 2y = f (x) , y(1) = 0 , y(2) + 2y\(2) = 0.
Formulate the corresponding left-end and right-end initial value problems. You
do not need to solve the IVPs.
(c) Assume that
yL (x) = 二 x2 and yR (x) =
are the solutions to the IVPs in part (b). Write down the Green’s function G(x, s) for the BVP in simplified form. [5]
(d) Represent the solution to the BVP in terms of the Green’s function G(x, s) for the particular choice f (x) = ex . You do not need to evaluate the resulting integrals. [5]
Question 5. [20 marks] Consider the autonomous dynamical system given by
x˙ = 4y , y˙ = 二x.
(a) Rewrite the system in matrix form and find the associated eigenvalues and eigenvectors.
(b) Determine the solutions of the corresponding initial value problems for the general initial conditions x(0) = a, y(0) = b.
(c) Sketch the phase portrait in the (x, y) phase plane and describe the shape of the
trajectories in the phase plane. If the initial condition is (x(0), y(0)) = (3, 4), describe the qualitative behaviour of the solution given in the phase portrait.
(d) Determine all fixed points of the system and describe the stability of
(x(t), y(t)) = (0, 0) as a solution for the linear system. What type of equilibrium point is (0, 0)?
2023-01-04