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 break down the two problems systematically and solve them step
by step.

Part (a): Solve the Partial Differential Equation (PDE)
uxy(x,y)=xy3u_{xy}(x, y) = xy^3uxy(x,y)=xy3 with the given
boundary conditions.

Problem:

We are given the PDE:

uxy(x,y)=xy3,forx,y≥0,u_{xy}(x, y) = xy^3, \quad \text{for} \quad x, y
\geq 0,uxy(x,y)=xy3,forx,y≥0,

with the boundary conditions:

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

Step 1: Integrating the PDE with respect to yyy