Optimize Matlab Derivative for Spacecraft Mission

[Juanka]

Homework Statement

Assume you have a spacecraft capable of a constant thrust (T) mission. The spacecraft ’s electric thruster has an efficiency given by Eff= 1-(((Isp-5000)^2)/(5000^2)) , where Isp is the specific impulse, in seconds. The power supply mass is given by
Mp=(alpha*g*T*Isp)/(2*Eff). The propellant mass is given by dM=Mo*(1-exp(-dV/(Isp*g))), where mo is the initial total spacecraft mass. dV=1.4*104m/s, g=9.81m/s2, T=0.3N, Mo=50000kg, and alpha=10kW/kg. Determine the optimum specific impulse, Iopt (where (Δm+mp) is at a minimum) for Isp =0..10000s.

(Hint: You will be taking the derivative of (dM+mp) with respect to Isp, then use optimization to determine Iopt.)

The Attempt at a Solution

First of all, I understand I have to add the dM and Mp equations together and take the derevidive respect to Isp. I think I should use the a form of the

simplify(dsolve('DIsp=dM+mp,'y(0)=?','Isp'))

However I do not know what my initial condition should be. Also I think I may have to solve the equations for Isp? am i on the right track and suggestions please.
Also do I wait until I have found the Derivative to input the given definitions of each variable or should I do that last?

 

[Juanka]

So I have been playing with code and I have came up with the following:

%% Given Definitions
dV=1.4*10^4; %units are in m/s
g=9.81;%units are in m/s^2
T=0.3;% units are in N
Mo=50000;% units are in kg
alpha= 10/1000; %units are in kg/W
%Isp=.1;
%% Given Equations
Eff=@(Isp) 1-(((Isp-5000)^2)/(5000^2)); 
Mp=@(Isp, Eff) (alpha*g*T*Isp)/(2*Eff);     
dM=@(Isp) Mo*(1-exp(-dV/(Isp*g)));
%% Analytical Soultion
syms Isp
diff((alpha*g*T*Isp)/(2*(1-(((Isp-5000)^2)/(5000^2)))) + Mo*(1-exp(-dV/(Isp*g))))

Am I on the right track?