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Question 1 (a) For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) x 2025 AU - U = ex+y. . sin x Uyx sin U = 0. (b) Consider the equation Ux - 2Ut = -3. (i) Find the characteristics of this equation. (ii) Find the general solutions of this equation. (iii) Solve the following initial value problem for this equation Ux-2U₁ = -3 |U(x, 0) = 0. - (c) Suppose V is a solution to the PDE V – VÃ = 0 and W is a solution to the PDE Wt+2Wx = 0. (i) Prove that both V and W are solutions to the following 2nd order PDE Utt Utx-2Uxx = 0. (ii) Find the general solutions to the 2nd order PDE (1) from part c(i). (d) Find the general solutions U(x, y) of the following equation Uxy = 1. (1) (e) Decide whether the following statements are true or false. (You don't need to explain your answer) (i) If U (x, t) is a solution to the heat equation Ut - Uxx = 0, then so is V(x,t) = U(x, −t). (ii) If U (r, 0) is a solution to the Laplace equation in the annulus N = {}/{ ≤r ≤ 2}, then U attains its maximum value at either the circle {r2} or the circle {r = ½}. (iii) If U solves the wave equation Utt - c²Uxx - 0 on the real line and has compactly supported initial data, then E[U](t) = }} S∞ [U² + c²U²²] satisfies ¿E[U](t) = 0, i.e. E[U](t) is a conserved quantity.

Question 1 (a) For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) x 2025 AU - U = ex+y. . sin x Uyx sin U = 0. (b) Consider the equation Ux - 2Ut = -3. (i) Find the characteristics of this equation. (ii) Find the general solutions of this equation. (iii) Solve the following initial value problem for this equation Ux-2U₁ = -3 |U(x, 0) = 0. - (c) Suppose V is a solution to the PDE V – VÃ = 0 and W is a solution to the PDE Wt+2Wx = 0. (i) Prove that both V and W are solutions to the following 2nd order PDE Utt Utx-2Uxx = 0. (ii) Find the general solutions to the 2nd order PDE (1) from part c(i). (d) Find the general solutions U(x, y) of the following equation Uxy = 1. (1) (e) Decide whether the following statements are true or false. (You don't need to explain your answer) (i) If U (x, t) is a solution to the heat equation Ut - Uxx = 0, then so is V(x,t) = U(x, −t). (ii) If U (r, 0) is a solution to the Laplace equation in the annulus N = {}/{ ≤r ≤ 2}, then U attains its maximum value at either the circle {r2} or the circle {r = ½}. (iii) If U solves the wave equation Utt - c²Uxx - 0 on the real line and has compactly supported initial data, then E[U](t) = }} S∞ [U² + c²U²²] satisfies ¿E[U](t) = 0, i.e. E[U](t) is a conserved quantity.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.5: Trigonometric Graphs
Problem 14E
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Question 1 (a) For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) x 2025 AU - U = ex+y. . sin x Uyx sin U = 0. (b) Consider the equation Ux - 2Ut = -3. (i) Find the characteristics of this equation. (ii) Find the general solutions of this equation. (iii) Solve the following initial value problem for this equation Ux-2U₁ = -3 |U(x, 0) = 0. - (c) Suppose V is a solution to the PDE V – VÃ = 0 and W is a solution to the PDE Wt+2Wx = 0. (i) Prove that both V and W are solutions to the following 2nd order PDE Utt Utx-2Uxx = 0. (ii) Find the general solutions to the 2nd order PDE (1) from part c(i). (d) Find the general solutions U(x, y) of the following equation Uxy = 1. (1) (e) Decide whether the following statements are true or false. (You don't need to explain your answer) (i) If U (x, t) is a solution to the heat equation Ut - Uxx = 0, then so is V(x,t) = U(x, −t). (ii) If U (r, 0) is a solution to the Laplace equation in the annulus N = {}/{ ≤r ≤ 2}, then U attains its maximum value at either the circle {r2} or the circle {r = ½}. (iii) If U solves the wave equation Utt - c²Uxx - 0 on the real line and has compactly supported initial data, then E[U](t) = }} S∞ [U² + c²U²²] satisfies ¿E[U](t) = 0, i.e. E[U](t) is a conserved quantity.
Transcribed Image Text:Question 1 (a) For each of the following equations, write down the order of the equation, determine whether each of them is linear or non-linear, and say whether they are homogeneous or inhomogeneous. (i) x 2025 AU - U = ex+y. . sin x Uyx sin U = 0. (b) Consider the equation Ux - 2Ut = -3. (i) Find the characteristics of this equation. (ii) Find the general solutions of this equation. (iii) Solve the following initial value problem for this equation Ux-2U₁ = -3 |U(x, 0) = 0. - (c) Suppose V is a solution to the PDE V – VÃ = 0 and W is a solution to the PDE Wt+2Wx = 0. (i) Prove that both V and W are solutions to the following 2nd order PDE Utt Utx-2Uxx = 0. (ii) Find the general solutions to the 2nd order PDE (1) from part c(i). (d) Find the general solutions U(x, y) of the following equation Uxy = 1. (1) (e) Decide whether the following statements are true or false. (You don't need to explain your answer) (i) If U (x, t) is a solution to the heat equation Ut - Uxx = 0, then so is V(x,t) = U(x, −t). (ii) If U (r, 0) is a solution to the Laplace equation in the annulus N = {}/{ ≤r ≤ 2}, then U attains its maximum value at either the circle {r2} or the circle {r = ½}. (iii) If U solves the wave equation Utt - c²Uxx - 0 on the real line and has compactly supported initial data, then E[U](t) = }} S∞ [U² + c²U²²] satisfies ¿E[U](t) = 0, i.e. E[U](t) is a conserved quantity.
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