APM3701 Assignment 1 (COMPLETE ANSWERS) 2025 (608471) - DUE 29 May 2025

apm3701 A+

ASSIGNMENT 1
SEMESTER 1
UNIQUE NO 608471




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, APM3701 Assignment 1 (COMPLETE ANSWERS) 2025
(608471) - DUE 29 May 2025; 100% TRUSTED
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QUESTION 1


Solve the following (initial)-boundary value problem, a. uxy (x, y) = xy3, x, y
0. u (x, 0) = f (x) , and uy (0, y) = g (y) . Determine u (x, y) , if f (x) = cosx
and g (y) = y+sin y. (Check your answer by substituting, and explain all the
steps clearly) (15 Marks) b. xux + yuy = yu u (2x2, x) = x2 − 1. (1) (Check
your answer by substituting before applying the boundary condition, and
explain all the steps clearly) (15 Marks) [30 Marks]



(a) Solve uxy(x,y)=xy3u_{xy}(x, y) = xy^3uxy(x,y)=xy3, with boundary
conditions:

u(x,0)=f(x)=cos⁡x,uy(0,y)=g(y)=y+sin⁡yu(x,0) = f(x) = \cos x, \quad u_y(0,y) =
g(y) = y + \sin yu(x,0)=f(x)=cosx,uy(0,y)=g(y)=y+siny



Step 1: Integrate with respect to yyy

∫uxy dy=∫xy3 dy\int u_{xy} \, dy = \int xy^3 \, dy∫uxydy=∫xy3dy
ux(x,y)=xy44+C1(x)u_x(x, y) = \frac{x y^4}{4} + C_1(x)ux(x,y)=4xy4+C1(x)