Plotting Differentials

Differential equations, anyone help with modeling a free fall question with drag?

Let the positive direction be down. Then from Newton's Law,

F = m dv/dt = mg - kv   (both before and after the chute opens)

where m is the mass of the skydiver, g is the acceleration due to gravity (32 ft/s²), (so mg is his weight), and k is the proportionality constant for the air resistance force (k=0.75 before the parachute opens, k=12 after it opens). We know from physical considerations that the skydiver's velocity will always be downward, so the air resistance force will always act upward; hence the minus sign (and absence of absolute value sign) for that term in the differential equation.

After dividing by m, we have

dv/dt = g - (k/m)v

For the free-fall part of the dive, the initial condition is v(0) = 0.

So solve the DE above, with k=0.75 and v(0) = 0. This will enable you to answer (a).

Integrate v(t) with respect to t to get s(t), the distance traveled as a function of time (with s(0) = 0). This will enable you to answer (b).

The limiting velocity is the velocity at which the air resistance force and the force of gravity are equal and opposite (so no net force):

kv = mg
v = mg/k

For (d), you will first need to solve the DE again, only with the different k, and the different "initial" condition (using your answer to (a)). Since the DE does not depend explicitly on t, you can "reset the clock", taking t=0 as the moment when the chute opens. As before, you will need to integrate this to get distance as a function of time. But to answer (d), you will need to set this distance function equal to (5000 - distance traveled during free fall), using your answer to (b), and solve for t. You will have to do this graphically or numerically; the equation won't be algebraically solvable for t.

Is differential equations or linear algebra hard?

I was enrolled in linear and diff eq at the same time. Dropped linear after a week because the teacher is notoriously bad. I thought people were over exaggerating but he was really that bad.

Anyways, my friend stayed in the class and was also taking diff eq with me. She got A's in both but is extremely intelligent.

Differential equations is pretty easy. At least I thought it was but i know plenty of people that were scraping by with c's. I got an A and didnt put in that much effort. I studied for exams and did a decent amount of homework. Like every other odd give or take depending on how well i understood the lesson or what i expected to see on an exam.

Took it along with with calc based physics electricity and magnetism, and computer science. Got a's in all of them. Just study for the exams, obviously give it more effort if you have a bit of a hassle. Depending on your schools emphasis on math you may want to make sure you understand proofs semi-well. They are not as crazy as linear algebra proofs. But our school's math classes are designed for math majors so there was a proof on every exam.

I took calc 3 before diff eq and we did not use any multivariable calculus stuff. Double integral popped up in proving convolutions but it was only in lecture.
Overall differential should be easy, combining it with linear is questionable. It depends on the rest of your course load. but its doable

Oh have you taken physics mechanics?
Our diff eq class loved to have harmonic oscilators, damped, undamped and forced harmonic oscilators.
We also went over RC circuits but only in one lesson

Direction fields? Differential Equations?

A direction field (or slope field) plot is just a plot of the slope (or, equivalently, the tangent lines) of a function y(t) at various points on the t-y plane. Here, you are given dy/dt = f(y,t). dy/dt, of course, is the slope of the function y(t), so all you need to do is evaluate dy/dt (= (4 - t*y)/(1 + y^2) over a grid of points on the t-y plane, and a short segment of a line with that slope at the gridpoint.

Alternatively, you can use one of the many direction-field plotting applets available on the web to do this. (See source, or do a web search on "direction field" or "slope field" plot.)

Solution Of Differential Equations using Euler's Method and Modified Euler's Method?

I have MATLAB on my computer and am an expert in it. I can give you a generic MATLAB code of solving differential equation using Euler's Method --

Here is a generic script that demonstrates Euler integration for a second order problem using Matlab.

% The problem to be solved is:

%y''(t)+100*y'(t)+1E4*y(t)=1E4 * abs(sin(wt))

h=0.05; %h is time step.
t=1:h:2; % initializing time variable.

clear x; %wiping out old variable.

x(1,:)=[0 0]; %initial condition is zero.
% x is 2D since there are 2 state variables.

for i=2:length(t), %Set up "for" loop.

%Calculating the values of k1 (note that it is a vector of length 2).
k1=[x(i-1,2) 1E4*abs(sin(377*t(i)))-100*x(i-1,2)-1E4*... derivatives;
x(i,:)=x(i-1,:)+k1*h; %Estimatingnew value of x;
end

%Plotting approximate and exact solution.
plot(t,x(:,1),'r',t,abs(sin(377*t)),'b'...
legend('Output','Input');
title('Euler Approximation, Second order system');
xlabel('Time');
ylabel('Input and Output');


You have to adjust and alter the functions and it should work well. I have already entered the step size and the interval [1,2].


Likewise I can type the modified Euler's Method MATLAB program too. It is similar to above. I hope that's helpful for you

Differential equation: Initial Value Problem: Oscillations and resonace: x''+4x'+5x=10cos(3t); x(0)=x'(0)=0

This problem asks to plot both the steady periodic solution Xsp(t)=C*cos(wt-(alpha)) of the differential equation and also the transient solution that satisfies the given initial solutions... any help you can give on how to solve or graph the problem would be much appreciated.