Mathematical Methods and Mechanics

Question I

Use the function of two independent variables x andy,

f (xy):i + 4ry-&+ 4y-9

(a) Find all the first and second order partial derivatives.

(b) Find the stationary poin(s) of the function,

(c) Classift the stationarypoin(s) of the function.

(4 marks)

(4 marks)

(4 marks)

Question 2

Consider the differential equation

$:t * *Ji, where 7>0, dx

with the initial conditiony =0.427. whenx= 0.3.

(a) Use Euler mid-point method with step size 0.01 to calculate an approximate value fory(0.31) accurate to 6 decimal places, whereflr) is the solution of the equation satisffing the given initial condition.

(5 marks)

(b) Using a computer program to calculate an approximation I(0.4), for y(0.4) by the Euler-trapezoidal method, the following values for I(0.4) were obtained for two different step sizes.

Steo size r0.4) 0.05 0.s51536 0.02 0.551521

Estimate the upper bound of the step size which would give an approximation to y(0.4) accurate to 6 decimal places, when used in the Euler-&apezoidal method.

(7 marks)