Some Differential equation problems

In summary, the conversation discusses two different issues. The first issue is finding the fundamental set of solutions for a given differential equation with a specific initial point. The second issue involves setting up a problem involving a spring and a damper. The person is requesting some hints on how to approach the problem and is not asking for a solution.
  • #1
seang
184
0
Hey all, I'm reviewing for an exam and I'm in need of a bump on the following issues:

1.) Find the fundamental set of solutions for the given differential equation and initial point.

y'' + y' - 2y = 0 tsub0 = 0

2.)A spring is stretched 10cm by a force of 3 N. A Mass of 2kg is hung from the spring and attached to a damper which exerts 3 N when the velocity = 5m/sec. There's more but I just need a little help setting it up. I don't understand how to find y (as in yu'(t)). Unless its just 3/5.

Just a few hints would suffice, I'm not asking you to solve these.

Thanks, Sean
 
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  • #2
for a) you have to assume that the solution looks like [tex] y(t) = Ce^{\lambda t} [/tex] where lambda is to be determined. Substitute this into your expression and you can find your lambda. Also use your initial condiotion to find C.
 
  • #3


1. To find the fundamental set of solutions for the given differential equation, first we need to solve the characteristic equation: r^2 + r - 2 = 0. This gives us two distinct roots, r = 1 and r = -2. Therefore, the general solution to the differential equation is y(t) = c1e^t + c2e^-2t.

To find the particular solution, we can use the initial point given (tsub0 = 0) to solve for the constants c1 and c2. Substituting t = 0 and y = 0 into the general solution, we get c1 + c2 = 0. Substituting t = 0 and y' = 0 into the general solution, we get c1 - 2c2 = 0. Solving these equations, we get c1 = 2 and c2 = -2. Therefore, the fundamental set of solutions is y(t) = 2e^t - 2e^-2t.

2. To find the equation for y(t), we can use Newton's second law: F = ma. In this case, the force (F) is equal to the sum of the spring force (kx, where k is the spring constant and x is the displacement) and the damping force (c(dx/dt), where c is the damping constant and dx/dt is the velocity). Therefore, we have the equation 3 = k(0.1) + c(5).

We also know that the mass (m) is equal to 2kg, so we can use the equation F = ma to solve for k. Substituting the values, we get k = 3/0.1 = 30 N/m.

To find the damping constant, we can use the given information that the damper exerts 3N when the velocity is 5m/sec. This means that c(5) = 3, so c = 3/5.

Therefore, the equation for y(t) is y(t) = (3/5)(dx/dt) + 30x.

I hope these hints help you in solving the problems. Good luck on your exam!
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It is commonly used to model dynamic systems in various fields of science and engineering.

2. How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding the exact form of the function, while numerical solutions use algorithms to approximate the solution.

3. What are initial value problems?

An initial value problem is a type of differential equation where the value of the function and its derivatives are known at a specific initial point. The goal is to find the solution for the function at other points.

4. What is the role of differential equations in science?

Differential equations play a crucial role in science as they are used to model and understand various natural phenomena, such as population growth, chemical reactions, motion, and many more.

5. Are there any real-world applications of differential equations?

Yes, differential equations have numerous real-world applications in fields such as physics, chemistry, biology, economics, and engineering. They are used to make predictions, optimize processes, and design systems.

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