The following is an initial value, second-order differential equation: d2 x dt 2 1 (5x) dx dt 1 (x 1 7)sin(vt) 5 0 where dx dt (0) 5 1.5 and x(0) 5 6 Note that v 5 1. Decompose the equation into two fi rst-order differential equations. After the decomposition, solve the system from t 5 0 to 15 and plot the results.

Aspacecraft can attain a stable orbit 300 kilometers above earth if it reaches a velocity of 7.7 kilometers per second. the formula for a rocket's maximum velocity v in kilometers per second is vequalsminus0.0098tplusc ln upper r, where t is the firing time in seconds, c is the velocity of the exhaust in kilometers per second, and r is the ratio of the mass of the rocket filled with fuel to the mass of the rocket without fuel. find the velocity of a spacecraft whose booster rocket has a mass ratio of 20, an exhaust velocity of 2.1 km/s, and a firing time of 15 s. can the spacecraft achieve a stable orbit 300 km above earth?