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

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APM3701
ASSIGNMEMT 1 SEMESTER 1
ENIQUE NO 608471




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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]




QUESTION 1

(a) Solve uxy(x,y)=xy3u_{xy}(x, y) = xy^3

Given:

 u(x,0)=f(x)=cos⁡xu(x,0) = f(x) = \cos x
 uy(0,y)=g(y)=y+sin⁡yu_y(0,y) = g(y) = y + \sin y

Step 1: Integrate with respect to xx

∫uxy dx=∫xy3 dx\int u_{xy} \,dx = \int xy^3 \,dx uy(x,y)=12x2y3+C(y)u_y(x,y) =
\frac{1}{2}x^2 y^3 + C(y)

where C(y)C(y) is an arbitrary function of yy.

Step 2: Apply boundary condition uy(0,y)=g(y)u_y(0,y) = g(y)

C(y)=y+sin⁡yC(y) = y + \sin y