This document is an exam for a differential equations course. It contains 4 problems to solve in 60 minutes. Problem 1 asks students to determine if functions are linearly dependent or independent on an interval. Problem 2 is an initial value problem to solve. Problem 3 is to solve a differential equation. Problem 4 is to find the general solution of another differential equation. Formulas for trigonometric integrals are provided.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
We have implemented a multiple precision ODE solver based on high-order fully implicit Runge-Kutta(IRK) methods. This ODE solver uses any order Gauss type formulas, and can be accelerated by using (1) MPFR as multiple precision floating-point arithmetic library, (2) real tridiagonalization supported in SPARK3, of linear equations to be solved in simplified Newton method as inner iteration, (3) mixed precision iterative refinement method\cite{mixed_prec_iterative_ref}, (4) parallelization with OpenMP, and (5) embedded formulas for IRK methods. In this talk, we describe the reason why we adopt such accelerations, and show the efficiency of the ODE solver through numerical experiments such as Kuramoto-Sivashinsky equation.
Question bank Engineering Mathematics- ii Mohammad Imran
its a very short Revision of complete syllabus with theoretical as well Numerical problems which are related to AKTU SEMESTER QUESTIONS, UPTU PREVIOUS QUESTIONS,
1. Concordia University March 20, 2009
Applied Ordinary Differential Equations
ENGR 213 - Section J
Prof. Alina Stancu
Exam II (A)
Directions: You have 60 minutes to solve the following 4 problems. You may use an admis-
sible calculator. No cell phones are allowed during the exam.
(1) (8 points) Determine whether the functions f1 (x) = x, f2 (x) = x2 , f3 (x) = 4x − 3x2
are linearly dependent or linearly independent on the interval (0, ∞).
(2) (15 points) Solve the initial value problem
y − 2y + 2y = 2x − 2, y(0) = 2, y (0) = 0.
(3) (12 points) Solve the differential equation
y + y = csc2 x.
(4) (5 points) Find a general solution of the differential equation
xy + 2y = 0.
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Useful Formulas: sec x = , csc x = , sec u du = ln | sec u + tan u| + C,
cos x sin x
csc u du = ln | csc u − cot u| + C.
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