**Question: **

(a) |
Consider the first order ODE
\[\frac{dy}{dx}={{e}^{-4x}}-3y,\,\,\,y(0)=1.\] |
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(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] |