Differential Equations: differential equations – #27260


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]




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