(a) Consider the first order ODE (dy)/(dx)={e^{-4x}}-3y, y(0)=1. (i)

Question:

(a)	Consider the first order ODE \[\frac{dy}{dx}={{e}^{-4x}}-3y,\,\,\,y(0)=1.\]
	(i)	Solve the above ODE above, using the integrating factor method to obtain an exact solution. Evaluate the exact solution at the point \[x=0.5\].	[5]
	(ii)	Use Euler’s method to obtain an approximate solution of the above ODE at \[x=0.5\], using \[{{x}_{0}}=0\] and step size \[h=0.1\]. Work to 3 decimal places.	[6]
(b)	A uniform circular hoop of mass 10 kg and radius 0.5 metres starts from rest and rolls, without slipping or falling over, and with its plane vertical, down a line of greatest slope of a fixed plane which is inclined at an angle \[{{45}^{o}}\] to the horizontal.
	(i)	Calculate the linear acceleration of the centre of the hoop.	[7]
	(i)	Find the linear velocity of the centre of the hoop once the hoop has rolled a distance of 1 metre.	[2]