Explanation Procedure used: Solution to a differential equation: The n th-order ordinary differential equation is one dependent variable by the general form, F ( x , y , y ' , ... , y ( n ) ) = 0 Where F is a real valued function of n + 2 variables: x , y , y ' , ... , y ( n ) . The solution of an n th-order ordinary differential equation is a function ϕ that possesses at least n derivatives and satisfying the differential equation on interval I , i.e. F ( x , ϕ ( x ) , ϕ ' ( x ) , ... , ϕ ( n ) ( x ) ) = 0 for all x in I . The Fundamental Theorem of Calculus states that the function F defined by, F = ∫ a x g ( t ) d t on the interval [ a , b ] , a ≤ x ≤ b , is differentiable on ( a , b ) and, F ′ ( x ) = d d x ∫ a x g ( t ) d t = g ( x ) Calculation: Consider the given differential equation d y d x − 2 x y = e x and the given function y = e x 2 ∫ 0 x e t − t 2 d t . The given function is a solution to the given differential equation if the function is satisfied by the differential equation

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

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Chapter 1, Problem 28CR

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To verify: The function y=ex2∫0xet−t2dt is a solution to the differential equation dydx−2xy=ex.

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Consider the cones K = = {(x1, x2, x3) | € R³ : X3 ≥√√√2x² + 3x² M = = {(21,22,23) (x1, x2, x3) Є R³: x3 > + 2 3 Prove that M = K*. Hint: Adapt the proof from the lecture notes for finding the dual of the Lorentz cone. Alternatively, prove the formula (AL)* = (AT)-¹L*, for any cone LC R³ and any 3 × 3 nonsingular matrix A with real entries, where AL = {Ax = R³ : x € L}, and apply it to the 3-dimensional Lorentz cone with an appropriately chosen matrix A.

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Let M = M₁U M₂ UM3 and K M₁ = {(x1, x2) ER²: 2 ≤ x ≤ 8, 2≤ x ≤8}, M₂ = {(x1, x2)™ € R² : 4 ≤ x₁ ≤ 6, 0 ≤ x2 ≤ 10}, M3 = {(x1, x2) Є R²: 0 ≤ x₁ ≤ 10, 4≤ x ≤ 6}, ¯ = cone {(1, 2), (1,3)†} ≤ R². (a) Determine the set E(M,K) of efficient points of M with respect to K. (b) Determine the set P(M, K) of properly efficient points of M with respect to K.

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To verify: The function y = e x 2 ∫ 0 x e t − t 2 d t is a solution to the differential equation d y d x − 2 x y = e x .

Solution Summary: The author explains how the function y=ex2displaystyle int is a solution to the differential equation.

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To verify: The function y=ex2∫0xet−t2dt is a solution to the differential equation dydx−2xy=ex.

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Chapter 1 Solutions