# Differential Equations: differential equations – #27258

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