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. S

    How Do You Calculate Mass Using Spring Constant and Frequency?

    Homework Statement (1) Suppose you have a spring with spring constant 4 N/m. You want to use it to weigh items. Assume no friction. You place the mass on the spring and put it in motion. a) You count and find that the frequency is 0.8 Hz (cycles per second). What is the mass? b) Find a...
  2. U

    Did I solve this diff eq substition problem properly?

    xy'=yln(xy) xdy=yln(xy)dx \frac{dy}{y}=\frac{ln(xy)dx}{x} Substitution: v=ln(xy) dv = \frac{dy}{y}-\frac{dx}{x} dv-\frac{dx}{x}=\frac{vdx}{x} ∫\frac{dv}{v+1}=∫\frac{dx}{x} ln(v+1)=ln x + C v+1 = Cx ln(xy) +1 = Cx Would that basically be the complete answer?
  3. G

    Graph theory vs. Diff Eq approaches towards studying complex networks

    Can anyone comment on the advantages and disadvantages of both graph theory vs. using a system of differential equations to study a complex network? For example, how much computing power and running time would a graph theory approach use compared to say solving a system of 100 differential...
  4. B

    How to solve a second order diff eq?

    If have this equation: \frac {d^2x}{dt^2}=-\frac{x}{1+x^2} How do I solve it?
  5. K

    Early Diff EQ Solving Methods (Chp 1 status yo)

    Alright, got a quick differential equation question. So far in my DE class, we have learned 3 methods of solving ODE's -Separation of Variables -"Guessing" (the y=yh+yp method) -Integrating Factors How do you know when to use these methods and when not to? I understand the methods to...
  6. N

    B in Diff Eq Honors or A in Non-Honors?

    Should I take Diff Eq honors which goes deeper and broader into the material and obviously has harder test problems and risk getting a B or C or take regular where I will probably get an A? What sucks is given enough time I can eventually figure out the hard honors problems but since tests have...
  7. G

    Physical Applications of the Bernoulli Diff Eq

    I am curious what the nonlinear bernoulli equation is used to model. Is there a certain topic or context where it shows up often? Can any suggest some references for more info? I am reviewing some ODE stuff for an upcoming exam and would really like an intuitive feel for the equation and its...
  8. B

    Laplace equation w/ dirichlet boundary conditions - Partial Diff Eq.

    Homework Statement The steady state temperature distribution T(x,y) in a flat metal sheet obeys the partial differential equation: \displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 Seperate the variables in this equation just like in the...
  9. S

    What kind of problem is this. (Seperable or Bernoullis) / Diff EQ

    Homework Statement xy^2 dy/dx = y^3 - x^3 , y(1) = 2Homework Equations The Attempt at a Solution It says to solve the initial value problem. I am assuming it is not a Bernoulli, but I can't seem to separate it. What should I do?Thanks This is what I get when I separate it, is this...
  10. L

    2nd order homogeneous diff eq

    Homework Statement y"-2y'+5=0, y(∏/2)=0, y'(∏/2)=2 find general solution of this diff eq Homework Equations The Attempt at a Solution i have followed all of the steps for this, rather easy 2nd order diff eq, but i my solution differs from the books solution. steps...
  11. C

    Diff Eq Jump Start: Calc II | Video Lectures & Notes

    I'm taking Calc II currently and trying to get a head start on diff eq. Does anyone know some good online resources that would assist me in this? Video lectures, notes, problems, ect... I'm very happy with Adrian Banner's calculus lectures and was seeing if there was anything else similar...
  12. B

    Drilled hole through earth - Diff EQ w/ Gauss' Law

    Homework Statement Suppose that a hole has been drilled through the center of the earth, and that an object is droppped into this hole. Write a first order linear differential equaiton for the object's velocity, v as a function of the distance r from the Earth's center (i.e, and equation...
  13. S

    A very theoretical approach to diff eq

    Hi. I'm taking diff eq course this semester and the text is the latest Boyce DiPrima diff eq with boundary value problems. The first test is mostly proofs on theorems about continuity, like the Heine-Borel theorem, Bolzano-Weierstrauss theorem, etc. The book doesn't go into much details about...
  14. L

    Solve 2nd Order Diff Eq: (-1/k^2)*(d^2y/dx^2)-y = (Q*c/P*L)*x

    Homework Statement Hi everyone, I have the following differential equation that I am trying to solve: (-1/k^{2})*(d^{2}y/dx^{2}) - y = (Q*c/P*L)*x Where Q,c,P,L, and k are constants. The solution ends up being: y = A*cos(k*x) +B*sin(k*x) -(Q*c/P*L)*x Where A and B are constants...
  15. C

    Specific solution of diff eq, no initial value

    Homework Statement Consider the differential equation Find a general solution to this differential equation that has the form . . . y = Cx3 Find a second solution that might not be a general solution and which may have a different value of n than your first solution. The Attempt at a...
  16. H

    Engineering Physics, Diff EQ, Matrix Algebra - too hard?

    Hello, I am currently signed up for Fall Quarter Engineering Physics, Diff EQ, and Matrix algebra at the University of Washington. Right now I am sort of skeptical if I should drop one of these classes because I have heard horror stories from at least one of each of these classes. I got a B-...
  17. G

    Diff Eq- Convolutions and state-free solution

    Homework Statement Find the state-free solution yii+4yi+29y=0 y2+4y+29=f(t) Homework Equations I know I have to find the impulse-response, e(t), which is 1/5e^-2t sin(5t). The Attempt at a Solution The state-free solutions is the convolution of e and f(t). This is the part I'm confused...
  18. S

    Courses Taking Honors Diff eq after shaky Calc III and Linear Alg courses

    Hi. I took Calc III and Linear alg over the summer. The course was 6 weeks and we did not cover everything in each chapter. From Calc III, we didn't cover things like curl, divergence, and Green's Theorem. In Linear Algebra, we didn't do orthogonal functions, diagonalization, Eigenvectors, or...
  19. ElijahRockers

    Non-homogenous Diff EQ, LRC circuit

    Homework Statement Consider an LRC circuit. L = 3/5, R = 10, C = 1/30, E(t) = 300, Q(0)=0, I(0)=0 i) Find charge and current. ii) Find maximum charge on the capacitor. Homework Equations LQ'' + RQ' +(1/C)Q = E The Attempt at a Solution (3/5)Q'' + 10Q' +30Q = 300 For the...
  20. ElijahRockers

    Equilibrium Solutions and General Form of Tsunami Model | Separable Diff EQ Work

    Homework Statement A model for the shape of a tsunami is given by \frac{dW}{dx} = W\sqrt{4-2W} where W(x) > 0 is the height of the wave expressed as a function of its position relative to a point off-shore. Find the equilibrium solutions, and find the general form of the equation. Use...
  21. ElijahRockers

    Diff EQ Direction field asymptotes

    Homework Statement Draw the direction field for the differential equation y'=1-y/x Homework Equations The Attempt at a Solution Ok well, drawing the direction field is not an issue because I have a grapher, and I get the basic of how to draw simple direction fields. So to...
  22. S

    Solving a diff eq by substituion

    Homework Statement x*y2(dy/dx)=y3-x3 solve the problem Homework Equations The Attempt at a Solution My answer was (x3/3)+c=y/x there is no answer in the back of the book for this one so i just want to make sure i got it right
  23. R

    Understanding First Order Linear Differential Equations: Solving and Simplifying

    Homework Statement The Attempt at a Solution I don't understand in step one why the three in the numerator disappears. I also don't understand why dy/dx becomes d/dx. the book just says left side is d/dx(v*y), a lot of help that is. how do you go to two fractions with x^3 and x^4...
  24. R

    Why Does dy Change in Solving Linear Differential Equations?

    Homework Statement I don't understand why in step 2 dy turns into d and why +P(x) dissappears I also don't see a difference between v(x)y and v(x)*y In step three why does the d disappear. I see that dx is going over to the right side, well, what about the numerator d?
  25. H

    Find particular solution third order Diff Eq

    Homework Statement y''' - y = e^x + 7 Homework Equations The Attempt at a Solution I used y=Ae^x +B and then I multiplied by x^2 because y_c = c1 + c2 e^x + c3 e^(-x) the c1 and c2 e^x value repeat. Therefore I got: y= Ax^2 e^x + Bx^2 I got A = 0 and A=1 which is wrong...
  26. J

    Diff Eq. Proving the existence of a unique solution

    Can somebody help me out here? Consider y'(t) = y(t)[a(t) - b(t)y(t)] where a,b : (-infinite, +infinite) → (0, +infinity) and there exists M>0 such that: (1/M) ≤ a(t), b(t) ≤ M, for all t in the reals Claim: There exists a unique positive solution Phi(t) defined for all t in the reals in...
  27. A

    2nd order homogeneous linear diff eq

    Homework Statement y'' + y' - 2y = 0 Homework Equations The Attempt at a Solution I think this is extremely simple. hopefully i am correct. i said the 'auxiliary' equation is r2 + r - 2 = (r+2)(r-1) = 0 the roots are r = 1, -2 so the solution is y=c1ex + c2e-2x correct?
  28. Totalderiv

    Diff Eq- Nonhomogeneous Equations

    Homework Statement Find a particular solution of the given equation. y^''' + 4y^' = 3x-1 Homework Equations r^3 + 4r = 0 r = 0, r = 2i, r = -2i The Attempt at a Solution y(x) = Ax-B y^'(x) = A y^''(x) = 0 y^'''(x) = 0 The answer is: y(x)=(3/8)x^2 - (1/4)x But I'm not...
  29. P

    How Does Opening a Switch Affect a Circuit with a Capacitor and Resistors?

    Homework Statement http://imgur.com/VsDrQ,dJ2iv Homework Equations Current in loop 1: i_1 going counter-clockwise Current in loop 2: i(t) going counter-clockwise Before opening switch: we know that 2 loops exist, left and right. also the current is constant because it says the...
  30. S

    What is the polar form of a complex number raised to a fractional power?

    Homework Statement I have some trouble understanding some of my math homework maybe somebody can help me out? 1) find all the values of ln(e) (-1)^i I know that i am going to have to use eulers formula in some way i believe but I am not really sure what the question is asking, what...
  31. V

    Perturbation theory to solve diff eq?

    Hi all, I have a tricky problem in pertubation theory. I have a function: f(\vec{r}) = P(\vec{r}) + \left( B(\vec{r}) + b(\vec{r}) \right)^2 where b(\vec{r}) is a small perturbation and is equal to 0 when P(\vec{r}) = 0 Now, to solve the equation \nabla f(\vec{r}) = 0 for b(r) is...
  32. C

    Brushing up on the basics of diff eq

    Homework Statement y(x) = C1sin(2x) + C2cos(2x) Homework Equations y(∏/8) = 0 The Attempt at a Solution C1(1/2)(√2) + C2(1/2)(√2) + 1 = 0 The book jumps to the equation below and I'm having trouble figuring out how they got there. C1 + C2 = -√2
  33. B

    Legendre's Diff Eq using Frobenius

    Homework Statement Show, by direct examination of the Frobenius series solution to Legendre's differential equation that; P_n(x) = \sum_{k=0}^{N} \frac{(-1)^k(2n-k)!} {2^n k! (n-k)! (n-2k)!}x^{n-2k} ;\ N=\frac{n}{2}\ \mathrm{n\ even,}\ N=\frac{n-1}{2}\ \mathrm{n\ odd} Write down the first...
  34. Z

    Finding Solution to 2nd Order Diff Eq at x=1

    Homework Statement Find the solution to the following 2nd Order Differential Equation at x = 1: y'' = 10y -200 Boundary conditions: when x = 0, y = 100 when x = 1, y' = 10 2. The attempt at a solution Complimentary function: y = A exp{(10)^0.5 x} + B exp{- (10)^0.5 x} Particular Integral: 20...
  35. G

    Initial value problem of ord diff eq

    Homework Statement 3y'' -y' + (x+1)y = 1 y(0) = y'(0) = 0 Homework Equations Not sure, that's the issue The Attempt at a Solution I can't quite get this one using the methods I'm familiar with, and I can't guess a particular solution to neither the equation nor the...
  36. C

    Good Theory-Intensive Introductory Diff Eq Textbook

    I am currently taking an introductory course to DE's that has a heavy emphasis on theory (my professor stresses that this is a math course, not an engineering/applications course). Unfortunately, my professor, though he tries hard, is not good at explaining anything. Even more unfortunately, the...
  37. O

    Finding general solution to a second order forced diff eq

    I am trying to solve the following problem and am a bit lost so any advice would be welcomed. x'' = 2x' + x = 3cos2t + sin2t My understanding is that I need to find the general solution for the unforced equation and a particular solution of the above equation. When these are added together...
  38. G

    Why doesn't this diff eq. have two solutions?

    Say you have the diff eq. x'=2x(x-13); x(0)=20. After separating and integrating we get, ln|(x-13)/x|=26t+C From here, we raise e to the power of each side to get rid of the natural log. Does this get rid of the absolute value signs? If so, why? If not, why is there only one solution (as...
  39. E

    Bacterial Growth: Finding the Initial Size and Doubling Period

    Homework Statement The count in a bacteria culture was 900 after 15 minutes and 1400 after 30 minutes. What was the initial size of the culture? What is the doubling period? What is the size after 70 minutes? When will the population reach 11000? Homework Equations P0= 900/e15k=1400/e30k...
  40. DocZaius

    Frequency of oscillating electric dipole in uniform field without using diff eq

    Homework Statement Electric dipole makes small angle with uniform electric field. Find the frequency of the oscillation using dipole moment p, moment of inertia I, and field magnitude E. Homework Equations Torque=I*(angular acceleration)=I*(theta)'' Torque=p*E*sin(theta) The...
  41. 1

    Can someone show me how to Diff Eq?

    Homework Statement I'm not taking a de class, just curious. dy/dx = (-5x+10y)/(9x) y(1) = -2 Use appropriate variable change to solve initial value problem. Homework Equations The Attempt at a Solution So, I'm given the derivative of function y, and want to know the...
  42. N

    Vector Equations to sys of diff eq

    Homework Statement The problem has four very similar parts: A)Rewrite the following vector equations as systems of differential equations: \frac{q}{A}=-k \nabla T (q is a vector) (spherical coordinates; k and A are constants) B)Rewrite the following vector equations as systems of...
  43. B

    What is the solution to the differential equation dy/dx = [10]x?

    Homework Statement Now that I'm working it out as I'm typing it, I think I may have solved this problem; could still be a mistake though... This was an extra credit problem on the final for my into to diff eq class. I never saw anything like it before and I didn't finish it because I got...
  44. M

    Ordinary/Partial Diff Eq books? And any introductions to Green's functions?

    I took a class on ODEs and have used them in my physics classes fairly often, but I would like a book that I could go back and learn more about differential equations. Tenenbaum and Braun have been suggested but there are quite a lot of books out there, searching has The second part is that...
  45. C

    How to Visualize Differential Equations?

    I don't feel that I can truly appreciate a math without being able to visualize it in my head. Generally speaking: calculus flows into areas, trig builds shapes, and linear algebra builds spaces, but I cannot for the life of me look at a diff eq and 'see' it, so to speak. While only proficient...
  46. A

    Solve second order diff eq with substitution

    Homework Statement Solve: xy''+2y'=12x^{2} with u=y' Homework Equations if you have: y'+P(x)y=Q(x) then your integrating factor is: I(x)=e^{\int P(x) dx} The Attempt at a Solution The only reason I was able to solve this is because I stumbled upon a...
  47. A

    Linear diff eq - correctly done?

    Homework Statement Solve the linear differential equation: xy'-2y=x^{2} Homework Equations If you have a linear differential equation of the form: y'+P(x)y=Q(x) then your integrating factor is: I(x)=e^{\int P(x) dx} The Attempt at a Solution If we divide both...
  48. A

    Seperable diff eq, differing intial-conditions

    Homework Statement Solve: \frac {dy}{dx} = 2x \sqrt{1-y^{2}} then find a solution for: y(0)=0 and can you find a solution for: y(0)=2 Homework Equations The Attempt at a Solution Just want to know if this is right. First the equation can be rearranged to...
  49. A

    Solve Seperable diff eq with substitution

    Homework Statement Solve the differential equation: x \frac {dy}{dx} = y + e^{\frac {y}{x}} with the change of variable: v = \frac {y}{x} Homework Equations The Attempt at a Solution I would just like to know if I have successfully solved the problem. Thanks...
  50. E

    Solving 2nd Order Diff Eq in Quantum Mechanics

    So I was trying a few quantum mechanics problems and encountered this diff eq: \frac{\hbar^2}{2m}\frac{\partial^2}{\partialx^2}\psi(x) + \frac{1}{2}kx^2\psi(x) = E\psi(x) I put it into the form: \frac{\partial^2}{\partialx^2}\psi(x) + (\frac{2mE}{\hbar^2} - \frac{m}{\hbar^2}kx^2)\psi(x) = 0...
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