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This is exam paper of Differential Equations course. It can be useful to engineering, computer science, physics and maths students. It was designed and taken by Prof. Dhanesh Bhatnagar at B R Ambedkar National Institute of Technology. It includes: Linear, Differential, Equation, Constant, Coefficients, Interval, Continuous, Variation, Parameters
Typology: Exams
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1. (a) (10 points) Find a linear differential equation with constant coefficients that has solutions t, et^ and tet^.
(b) (10 points) Prove or disprove that t^4 and t^6 can be solutions of one and the same linear ho mogeneous differential equation y��^ + p(t)y�^ + q(t)y = 0 on an interval [− 1 , 1], where p and q are continuous on [− 1 , 1].
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2. For a certain regular linear differential operator L a basis of solutions of Ly = 0 is given by
y 1 (t) = e^2 t^ , y 2 (t) = 2t^2 + 2t + 1.
(a) (10 points) Compute the Wronskian of y 1 and y 2.
(b) (10 points) Using variation of parameters, find a solution of the differential equation Ly = t^2 e^2 t^.
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4. (20 points) Find the general solution of
y���^ + 3y��^ + 3y�^ + y = te−t^.
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5. Consider the initial value problem
y�^ = 1 + y 2 , y(0) = 0.
(a) (10 points) Using Picard’s iteration method obtain the iterates y 1 (t) and y 2 (t).
(b) (10 points) Show that the initial value problem has at most one solution in any interval of the form t ∈ (−a, a).
(c) (extra credits) Find the exact solution y(t) and show that limn→∞ yn(t) = y(t).
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