Differential Equations: differential equations – #27258


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 \[v(0)=5\text{ m}{{\text{s}}^{-1}}.\]
(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]

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