**Question: **

B1. |
(a) |
Solve the second order ordinary differential equation
\[\frac{{{d}^{2}}x}{d{{t}^{2}}}+\frac{dx}{dt}-2x=2t,\]
given the initial conditions \[x(0)=1,\,\,\,x'(0)=-1.\] |
[10] | |

(b) |
A body of mass 10 kg moves on a horizontal surface, subject to a resistance force of \[R(v)\] Newtons, where v is the instantaneous velocity and the initial velocity is \[v(0)=5\text{ m}{{\text{s}}^{-1}}.\] |
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(i) | Let the resistance force be \[R(v)=v+2{{v}^{2}}\]. Calculate the displacement of the body when it comes to rest. | [5] | ||

(ii) |
Suppose the resistance force is changed to \[R(v)=2+0.2{{v}^{3}}.\] Then the distance, x, travelled by the body from its initial position is
\[x=10\int\limits_{v}^{5}{\frac{v}{2+0.2{{v}^{3}}}}dv.\]
Use Simpson’s Rule with four strips to calculate the approximate value of x when \[v=2\text{ m}{{\text{s}}^{-1}}.\] Work to 4 decimal places in all your calculations. |
[5] |