# What is Diff eq: Definition and 271 Discussions

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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1. ### Finding motion where the acceleration depends on position and time

I have computed that the acceleration in my problem is a(t) = -gj - k/m(|r(t)| - L_0) * r(t)/|r(t)| Where a(t) is the acceleration vector, g is the gravitational acceleration, j is the unit vector in y-direction, k is the spring constant, m is the mass, r(t) is the position vector, |r(t)| is...
2. ### B Is there an analytical solution to this equation?

So I've derived the rocket equation in empty space and with constant gravity. Now I am interested in adding air resistance. I'm aware that there are 2 different models as if 0<Re<1 then F_drag=k*v and if 1000<Re<30000 then F_drag=1/2*A*rho*CD*v^2. And for my purpose the second model is most...
3. ### Green's functions (Fourier Series)

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: $$\tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx$$ So I have: -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...
4. ### I Quick question for Finding EOM with diff eq

I have been going through my old books again, and found myself a little stuck. I am not entirely sure if this would be better in this one or diffy eq. The problem starts with having you find equation of motions when a= -bv, where b is constant and v = v(t) Using method of separable equations...
5. ### Dimensional analysis of an equation of motion

Homework Statement The evolution of the density in a system of attractive spheres can be described by the following dynamic equation. $$\frac{\partial}{\partial t} \rho (r,t) = D_o [\nabla^2 \rho (r,t) + \beta \nabla \rho (r,t) \int dr' [\nabla V (|r-r'|)] \rho (r',t) g(r,r',t)]$$ a)...
6. ### Elliptic functions, diff eq, why proof on open disc holds for C

Homework Statement Hi I am looking at this derivation of differential equation satisfied by ##\phi(z)##. To start with, I know that such a disc ##D## described in the derivation can always be found because earlier in the lecture notes we proved that their exists an ##inf=min \omega ## for...
7. ### Critical points of a diff eq system

Homework Statement I have to calculate the critical points of the following system. $$x'=cx+10x^2$$ $$y'=x-2y$$ The Attempt at a Solution So I solve the system $$cx+10x^2=0$$ $$x-2y=0$$ So if $$x=2y$$ I have $$2yc+10*4y^2=2yc+40y^2=y(2c+40y)=0$$ and I get $$y=0$$ and $$y=-\frac{c}{20}$$ f I...
8. ### A Can I linearize this equation?

By using the laplace transform: $f(t)=sin(Φ(t))$ I want it in the form: F(S)/Φ(S) The purpose is to linearize it in order to put it into a larger transfer function, so far my only solution is to simplify it using taylor expansion.
9. ### A Need help solving Darboux equation

I'm working on a personal math project and I'm running into this system of differential equations. I have seen references which state the solutions are in terms of Hermite modular elliptic functions, but I do not know what those functions are. All of the references I can find on this equation...
10. ### Looking for book suggestions: see inside.... (physics and diff EQ)

So I'm ordering some supplemental books to guide my education. I'm having a difficult time with physics and diff EQ this semester and am generally unhappy with how things are going in these classes. I'm looking for a few suggestions. I saw on another post that the following book is very good for...
11. ### I What exactly is linear dependency? 2nd and 3rd Order Diff EQ

We are studying 2nd and 3rd order differential equations in class, and have touched on superposition and were talking about an equation being linearly dependent or independent. I received some good explanations from tutors about this, using vectors as examples, but I'm still a bit unclear on the...
12. ### Diff eq with constants... Eulers identity...

Homework Statement Find the general solution of the second order DE. y'' + 9y = 0 Homework EquationsThe Attempt at a Solution Problem is straight forward I just don't get why my answer is different than the books. So you get m^2 + 9 = 0 m = 3i and m = -3i so the general solution...
13. ### Diff eq. Interval of definition

Homework Statement In problems 11-14 verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. Question 13: y'' + y = tanx ; y = -cos(x)ln(sec(x) + tan(x)) Anyways, next problem... In...
14. ### Laplace transform (translation on the s-axis)

Homework Statement L-1{[(2s-1)]/[(s^2)(s+1)^3]} Homework Equations L{f(t)e^(at)}=F(s-a) The Attempt at a Solution I have tried million ways but the different exponents in the denominator are throwing me off. The other problem is that I cannot use partial fractions, the homework instructions...
15. M

### Courses Mechanics / E&M / Multivariable Calc / Diff EQ Self Study

Hello Physics Forums! I'm in a little bit of an interesting situation. I've completed the classes offered at my school, but have been told that if I can directed-study these four courses they will provide me with a credit bearing exam for college undergraduate credit. Do you have any...
16. ### Increasing the concentration of a solution - Diff Eq?

I have a question regarding a problem in chemistry. I ultimately solved it, but I wanted to know if the problem itself could be represented as a differential equation, and if the solution I found is the solution to that equation. The explanation is a little long, but I don't think I can make it...
17. ### A Test if 2nd order diff eq. can be derived from a Hamiltonian

Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is). Are there any ways to test whether or not the given...
18. ### Determining if Systems are Linear

Homework Statement For each of the following, determine if the system is linear. If not, clearly state why not. (a) ##y''(t)+15y'(t)+sin(y(t)))=u(t)## (b) ##y''(t)-y'(t)+3y(t)=u'(t)+u(t)## (c) ##y'(t)=u(t)## and ##z'(t)=u(t)-z(t)-y(t)## Homework Equations None The Attempt at a Solution...
19. ### Studying Why do I keep failing this in particular? (Differential Equations)

Hello everyone. I'm an undergrad physics major with one semester left and I'm having some trouble. I took off 3 years to work on my depression and came back last spring to finish my senior year. Now, before I left I was struggling in all my classes due to my depression, but one was worse...
20. ### Engineering RLC Circuit, Find Diff Eq for VC - Can someone check my work?

Homework Statement Find the differential equation for VC Circuit #1 Circuit #2 Homework Equations KCL The Attempt at a Solution Circuit #1: Node VC: ##\frac{V_C-V_{in}}{R_1}+\frac{1}{L_1}\int_{0}^{t}(V_C-V_2)+C_1\dot{V_C}=0## Node V2: ##\frac{1}{L_1}\int_{0}^{t}(V_2-V_C)+\frac{V_2}{R_2}=0##...
21. ### 2nd Diff eq application - Vertical stretch

Hello guys, first I am sorry I didn't use the template. I deleted it by mistake and I didn't know how to create a new thread from scratch. so I have two cases of the same problem, on with initial conditions of pulling down and the other pushing up, I tried different signs but It didn't work...
22. ### Diff eq - LT Discontinuous Sources HW

Homework Statement Homework Equations The Attempt at a Solution this is my attempt, I figured this would be easier than typing. any how, I find partial fraction method is so time consuming and very algebraically complicated, is there a trick or another method I could use to make this...
23. ### Ambiguity in the method applied for differential equations

Homework Statement Why do we need two solutions to solve a 2nd order linear differential equation? lets consider a differential equation with equal roots for auxiliary equation. So the reasoning behind why can't we use y=Aen1x+Ben2x as its general solution is because since the roots are equal...
24. ### I Is this general solution correct?

Question : General solution of dy/dt = -ay + b My solution : dy/dt = -a(y-b/a) (dy/dt)/(y-(b/a)) = -a Integrating both sides : ln | y-(b/a) | = -at + C e(-at+C) = y-(b/a) Ce(-at) = y-(b/a)
25. ### Need help solving a differential equation

Are there any known analytical method to solve the equation $$A\frac{d^2f(x)}{dx^2}+B\frac{df(x)}{dx}+Ce^{igx}f(x) = 0\hspace{1cm}?$$ All quantities appearing in that equation are complex except for ##g## and ##x##.
26. ### Elliptic partial differential equation

Hey guys, so my professor told me to take a look at an equation, because he thinks that there is a mistake. We are basically talking about exercise 6.3 (on last image). The pictures will show you the text, so that you have all the information, that I have http://puu.sh/mrNDl/ec19cdff63.png...
27. ### General Solution from Particular Solution

Just a question about the theory of solutions to differential equations? Given a second order differential equation and two particular solutions y1 and y2, what is the best way to find the general solution? i.e variation of parameters or something else
28. ### Need a refresher: 1st order linear diff eq

I thought I understood how to solve these sorts of equations, but apparently not.. 1. Homework Statement In Linear Algebra I'm solving diff eqs with eigenvectors to get all the combinations that will solve for a diff eq. The text then asked me to check my answer by going back and solving...
29. ### To take DiFF EQ vs. PDE (sequence)

Greetings all, I am registering for spring 2016 courses and have one question. I can pick up a math course and I have the option between two courses: 430 Formal Logic vs. 481 Applied Partial Differential Equations. I am a math and physics double major. Course list and description...
30. ### Friction constant minimizing the duration of vertical motion

Homework Statement The mass of a car that acts on one wheel is 100 kg. The elasticity (spring) constant in the suspension system of that wheel is k = 10^4N/m. Design the strut (find the friction/resistance constant c) such that any vertical motion of the wheel (set up for example by going over...
31. ### Learning Differential equations, week 2 level material

Homework Statement Find the general solution to the equation. Homework Equations (dy/dx) - y - e^3x=0 The Attempt at a Solution [/B] I rewrote this as dy/dx - y = e^3x This is a linear first order ODE, in the form dy/dx + P(x)y = f(x) P(x) = 1; f(x) = e^3x The integrating factor =...
32. ### Exploring Differential Equations in Astrophysics: A Project Idea

Homework Statement [/B] Hello all, I am looking more for an idea than an actual problem to solve. (That comes later.) Homework Equations I am starting a Differential Equations class, and I am looking for an idea for a project. We are to complete an application assignment in which we have to...
33. ### Using analog computer to solve 2nd-order diff eq

I'm trying to build a circuit to solve the differential equation x''+2x'+x = f(t), where f(t) is a sine wave with frequency 5Hz and amplitude 0.5V. I am supposed to get a sine/cosine wave (as the diff. eq is just the same as the ove governing spring-mass forced oscillation) as solution, but...
34. ### Best Guess for Partial Solution for Diff EQ?

Homework Statement Find the specific solution for: y''-2y'+y=xe^x+4, y(0)=1, y'(0)=1. Homework Equations N/A The Attempt at a Solution Since xe^x is already in the general solution of the homogeneous version of this diff eq (complementary solution), my first guess for a partial solution...
35. ### Non-autonomous diff eq and periodicity

Homework Statement Consider the first-order non-autonomous equation ##x' = p(t) x##, where ##p(t) ## is differentiable and periodic with period ##T##. Prove that all solutions of this equation are period with period ##T## if and only if ##\int_0^T p(s) ds = 0##. Homework EquationsThe Attempt...
36. ### Need some Help with some Partial Diff Eq

[b] Homework Equations N/a The Attempt at a Solution I am wondering if this is the right direction in solving this. any input would be great! Sorry for the messy handwriting!
37. ### Convergence of perturbative solutions to a non-linear diff eq

Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it. We have the equation $$\partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right)$$ We will look for solutions satisfying...
38. ### What are the general forms of implicit and parametric equations?

I was thinking about some form of classify all kinds of equations and system of equations. And the better classification that I found is by 1st to classify if the equation represents a curve or a surface. Represents a curve if the spatial coordinates are functions of an unique variable and...
39. ### Multivariable Calculus or Diff EQ over the summer?

Hi everyone, this is my first post and so I just want to say thank you in advance for any responses to my question. I recently applied for an internship which will take place over the summer of 2014 and it looks like I have a good chance at actually being accepted for the program. However, I...
40. ### Need Opinions: Intro to Diff Eq or Statics

Good morning everyone. I need some guidance regarding which course to take next semester. For some background, I am active duty Navy with three kids majoring in Mechanical Engineering. Due to my work schedule, I can only take 2 classes per semester (for now). I am already locked into Physics...
41. ### Diff Eq Question - Impulse Functions

Homework Statement From Boyce and DiPrima's Elementary Differential Equations (9th Ed.), Section 6.5, Problem 13: Consider again the system in Example 1: 2y'' + y' + 2y = \delta(t-5); y(0)=y'(0)=0 (Solution in text...
42. ### Solve Diff Eq: Is it Homogenous?

Homework Statement determine whether the given differential equations are homogenous and, if so, solve them. Homework Equations dy/dx = (( x^2 -2y^2)/xy ) The Attempt at a Solution i assumed it was homoG. Then i replaced ( x*v) for each Y and made dy/dx --> v+x*(dv/dx)...
43. ### Solving Diff EQ: y'=2cos^2x-sin^2x+y^2 w/y(0)=2

Homework Statement For the following differential equation: dy/dx = \frac{2cos^2x-sin^2x+y^2}{2 cosx} , -pi/2 < x < pi/2 show that the substitution y(x)=sin x + 1/u(x) yeilds the differential equation for u(x), du/dx = -u tan x - \frac{1}{2}sec x Hence find the solution y(x) to...
44. ### Finish this problem? (Diff Eq)

Homework Statement \frac{dx}{dt}=x-x^{2}Homework Equations The Attempt at a Solution I think the only thing I have wrong so far is how to finish it because I can't find anything wrong with my work, but I don't know how the book gets their final answer. Separate variables... \int...
45. ### Graphing diff eq slope field with TI Nspire CX CAS HELP please ?

Graphing diff eq slope field with TI Nspire CX CAS HELP please!? Every time I try to graph a slope field an error message pops up saying 'wrong number of initial conditions'. What does this mean? I tried graphing equations like dy/dt = t^(2) + t or dy/dt = 1-2y but I keep getting that dumb...
46. ### Find 'k' for x(t)=k is solution for diff eq

Homework Statement Find the value of k for which the constant function x(t)=k is a solution of the differential equation 6t^5dx/dt+7x−4=0. The Attempt at a Solution I have tried this several ways but no luck, here is one of my attempts, solve for dx/dt, dx/dt = (4-7x)/6t^5 take the...
47. ### Diff EQ - Imaginary identities

Homework Statement Determine the general solution of: y(6) + y''' = t The Attempt at a Solution Ok, r = 0, 0, 0, 1/2 +- 3i/√2, -1/2 + 3i/√2 What do I do with that last r value? It turns into ce-t somehow, but I don't see it. edit: typed a number in wrong, fixed now~
48. ### Runge-Kutta 4th order method (RK4) for second order Diff Eq

Homework Statement Hello, maybe this is due to my lack of understand of RK4, but I have an equation: x'' + b^2*x=0 (derivatives with respect to variable t) and I need to use RK4 to find the solution on an interval. I can readily find solutions analytically, but my understanding of RK4...
49. ### Water tower/spring system Diff Eq

Homework Statement Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support acts as the spring, where the induced vibrations are horizontal. Suppose that...
50. ### Showing that some functions are solutions to a diff eq.

Homework Statement Show that y1(t) and y2(t) are solutions to a certain differential equation. Homework Equations The Attempt at a Solution I plug both of these into my diff. eq. and I got the same thing. How is this showing that they are solutions to a diff. eq.? What if I got a different...