APM3701 Assignment 1 (COMPLETE ANSWERS) 2025 (608471) - DUE 29 May 2025; 100% TRUSTED Complete, trusted solutions and explanations.

, APM3701 Assignment 1 (COMPLETE ANSWERS) 2025 (608471) -
DUE 29 May 2025; 100% TRUSTED Complete, trusted solutions
and explanations.
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]


Let's work through the two parts of this problem step by step, explaining
each step clearly.



Part (a): Solve the boundary value problem
Given: The partial differential equation is:

uxy(x,y)=xy3,x,y>0u_{xy}(x, y) = xy^3, \quad x, y > 0uxy
(x,y)=xy3,x,y>0

with boundary conditions:

u(x,0)=f(x)=cos⁡xanduy(0,y)=g(y)=y+sin⁡y.u(x, 0) = f(x) = \cos x
\quad \text{and} \quad u_y(0, y) = g(y) = y + \sin
y.u(x,0)=f(x)=cosxanduy(0,y)=g(y)=y+siny.

Goal: We want to find u(x,y)u(x, y)u(x,y).



Step 1: Solve the PDE uxy(x,y)=xy3u_{xy}(x, y) = xy^3uxy(x,y)=xy3