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]