ASSIGNMENT 1: SEP 291: ENGINEERING MODELLING 1. Solve x y e dx dy 3 2 , y(0) 5 2. Solve y x dx dy dx d y 2 3 3.125 sin 2 2 , (0) 5, (x 0) 3 dx dy y 3. Solve y e x dx dy dx d y x 2 5 3.125 sin 2 2 , (0) 5, (x 0) 3 dx dy y 4. Solve the differential equations i) y´= A sin x ii) xy´ = Ax + y where A = Summation of your Student ID, e.g. if your Student ID is 212407299, then A will be 2+1+2+4+0+7+2+9+9=36. In i), give the particular solutions satisfying the condition y(0) = 1. In ii) give the solution satisfying y (1) = 0. 5. Solve the following boundary value problems i) y´´+ 4y´+ 13y = 0 y(0) = 0 y( ) /2 = 1 ii) y´´– 4y´+ 4y = 0 y(0) = 0 y(1) = 1 6. Find the solutions to each of the following second order equations, with the specified conditions. Remember to apply the conditions to the full solution – CF + PI. i)y´´+ 4y´+ 3y = 2ex y(0) = 0 y´(0) = 1 ii)y´´+ 4y = x + 1 y(0) = 0 y( 4 ) = 1 4 7. Find the general solution of the differential equations y´= f(x, y) where f(x, y) is given by i) 2x 5 – sin y ii)(x – y)/x iii) 3(y2 – 3y + 2) 8. Solve 4y´´+36y = cos3x using variation of parameters. 9. Solve the initial value problem y( ) 0, y ( ) 2 4 10sin , / // y y x x 10. Given y x x dx dy 2cos x 4 sin sin2 find y x( ), given that 3 y(0) (ie y 0 when 3 x ). 11. Find general solution using variation of parameters: y´´- 4y+4y= (x + 1) e2x 12. Given that y1 = x2 is a solution of x2y’’- 3xy’ + 4y=0, use reduction of order to find a second solution y2. 13. In an RL circuit, the differential equation formed using kirchoff’s law, is Ri + L di/dt =V. Solve this, using separation of variables, given that R= 10 ohm, L=3H and V=50 volts, and i(0) = 0. 14. A cup of coffee (temperature = 220°F) is placed in a room whose temperature is 70°F. After seven minutes, the temperature of the coffee has dropped to 150°F. How many more minutes must elapse before the temperature of the coffee is 130°F? 15. At time t = 0, a tank contains 4 lb of salt dissolved in 100 gal of water. Suppose that brine containing 2 lb of salt per gallon of brine is allowed to enter the tank at a rate of 5 gal/min and that the mixed solution is drained from the tank at the same rate. Find the amount of salt in the tank after 10 minutes.
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