What is Diffeq: Definition and 66 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. A

    I Can we solve a non-autonomous diffeq via Taylor series?

    I've occasionally seen examples where autonomous ODE are solved via a power series. I'm wondering: can you also find a Taylor series solution for a non-autonomous case, like ##y'(t) = f(t)y(t)##?
  2. Feodalherren

    Simple Linear DiffEq, not understanding the book

    The way I want to solve it is the way that I always want to solve separable linear diffEqs: after some trivial algebra and an easy integral I end up with t = (-1/2) ln (20-2x) +C Easy enough, solve for x(t) yields x(t) = 10 - (1/2)e^(-2t) + C Solve for C when x(0) = 3 yields C = -13/2 But...
  3. N

    I Problem when evaluating bounds....Is the result 1 or 0^0?

    Consider the summation ∑,i=0,n (t^(n-i))*e^(-st) evaluated from zero to infinity. You could break down the sum into: (t^(n))*e + (t^(n-1))*e + (t^(n-1))*e + ... + (t^(n-n))*e ; where e = e^(-st) To evaluate this, notice that all terms will go to zero when evaluated at infinity However, when...
  4. Steven Reichman

    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...
  5. M

    I Which derivatives should I review for my DiffEq course?

    Background: It has been about a year and a half since I took Calc 3 so I am not as familiar with using derivatives as I would like to be. Basically my math dept. had a concentration in math-stats that didn't even require differential equations at all, so I wasn't expecting to take the course...
  6. D

    Solving functions for S in a q-q* Hamilton-Jacobi diffeq

    Homework Statement Homework EquationsThe Attempt at a Solution So far I have a solution for a) as For b) I formulate the equation as and so far for c) I have My main idea at the moment is that as the Lagrangian was not time dependent, the Hamiltonian will not be. Following on maybe...
  7. grandpa2390

    How did my professor get from here to here? DiffEq

    Homework Statement X(x)=(Ae^kx+Be^-kx) Y(y)=(Csin(ky)+D(cos(ky)) V(x,y)=(Ae^kx+Be^-kx)(Csin(ky)+D(cos(ky))Homework Equations separation of variables The Attempt at a Solution so our boundary condition says that as x->infinity , V=0 this is only possible if A=0 so A=0 and Be^-kx= 0 so X=...
  8. Q

    DiffEq, Binomial Expansion and limits

    Homework Statement Use algebra to show that U(x) = −√x − 1 and L(x) = −√x satisfy the ’funnel condition’ U(x) − L(x) → 0 as x → ∞ Homework Equations Funnel condition: The two fences come together asymptotically, i.e. U(x) − L(x) is small for large x. The Attempt at a Solution I think that...
  9. Q

    DiffEQ, Snowing Rates, Distance Traveled

    Homework Statement Early one morning it starts to snow. At 7AM a snowplow sets off to clear the road. By 8AM, it has gone 2 miles. It takes an additional 2 hours for the plow to go another 2 miles. Let t = 0 when it begins to snow, let x denote the distance traveled by the plow at time t...
  10. davidbenari

    Failure to see the validity of an approximation to DiffEq.

    The following comes from Griffiths Intro. to QM (2nd Ed) page 53. We want to solve the Schrödinger Equation for the harmonic oscillator case using a power series method. The details aren't important but you want to solve ##h''(y)-2yh'(y)+(K-1)h=0## whose recursion formula is...
  11. B

    How do I solve these coupled Differential Equation?

    Homework Statement dNa/dt = -Na/Ta where Na is the function and Ta is the constant dNb/dt = Na/Ta - Nb/Tb where Nb is the function and Tb is the constant Homework Equations My Prof said Nb(t) has the form Nb(t) = Cexp(-t/Ta) + Dexp(-t/Tb) The Attempt at a Solution I know the first equation...
  12. R

    Engineering LC circuit (Differential Equation)

    Homework Statement An inductor with value L and a capacitor with value C are connected in series to a power source. At time t, the voltage of the power source (i.e. the voltage across both the inductor and capacitor) is given by ## v(t)=Asin(\frac{2t}{\sqrt{LC}}) ##. If the voltage across the...
  13. Adoniram

    Equation of motion: Help with DiffEq (2nd order non linear)

    I am trying to solve the differential equation that will give me the equation of motion of a point charge under the influence of another point charge's electric field. Say point charge A is free to move, and it currently a distance D away from point charge B. Point charge B is fixed in space...
  14. T

    MHB What is the decay constant of tritium?

    "Tritium is the basic fuel of hydrogen bombs and is used to increase the power for fission bombs . . . In a basic atom bomb (or reactor), plutonium atoms are split, or fissioned, to release energy, but the fission can be promoted with a small amount of tritium because it has two extra...
  15. C

    Numerical Solution to Complex DiffEQ?

    I've been trying to figure out a way to get an approximation to a complex DiffEQ. dx/dt = c1 / (c2 + c3*x*t) Does anyone have any input on wether this problem can be approximated? Thank you.
  16. sheldonrocks97

    Find constants that satisfy integrals?

    Homework Statement ∫y1(x)^2dx from - to + infinity=1 and ∫y2(x)^2dx from - to + infinity=1 Homework Equations None that I know of. The Attempt at a Solution I evaluated the integrals and got that c1 is equal to c2 but I think that's wrong.
  17. _N3WTON_

    Solving Euler's Equation: Finding Solutions of the Form x^r

    Homework Statement Part 1: Show that Euler's equation has solutions of the form x^r. This can be found by obtaining the characteristic equation ar(r-1) + br + c = 0 Part 2: Solve the following Euler equation: x^{2}y'' + xy' = 0 Homework Equations Euler's Equation: ax^{2}y'' + bxy' + cy = 0...
  18. _N3WTON_

    The Principle of Superposition for Homogeneous Equations (DiffEq)

    Homework Statement Verify that e^x and e^-x and any linear combination c_1e^x + c_2e^{-x} are all solutions of the differential equation: y'' - y = 0 Show that the hyperbolic sine and cosine functions, sinhx and coshx are also solutions Homework Equations Principle of Superposition for...
  19. _N3WTON_

    Show the relation is an implicit solution of the DiffEQ

    Homework Statement Differential equation: 2xyy' = x^2 + y^2 Relation: y^2 = x^2 - cx Homework Equations The Attempt at a Solution Hello, I can normally solve this problems with ease; however, I am having trouble with this particular problem. I have performed the implicit...
  20. vanceEE

    Is this the correct solution to this DiffEq?

    $$ xy' = y + xy^2 $$ $$ xy' - y = xy^2 $$ $$ y' + (-1/x)y = y^2 $$ $$ e^{∫(-1/x)dx} = 1/x $$ $$ (1/x)[y' + (-1/x)y = y^2] $$ $$ (1/x)y' - (1/x^2)y = (1/x)y^2 $$ $$ ∫((1/x)y)' = ∫((1/x)y^2) dx $$ $$ (1/x)y = (y^2)ln|x| + C $$ $$ y = (xy^2)ln|x|+Cx $$ Wolfram shows me another solution... $$y =...
  21. vanceEE

    Solve Ordinary Differential Equation: ln(y)=xy

    I've come across this problem while self studying Ordinary Differential Equations and I really need help. The problem asks me to simply eliminate derivatives, I do not need to separate. The book shows the answer, but not the steps. problem: \frac{dy}{dx}= \frac{y^2}{1-xy} answer...
  22. F

    Series solution to diffeq, stuck on matching the indices

    Homework Statement find the series solution to y''+x^2*y'+y=0 Homework Equations y=summation from n=0 to infinity Cn*x^n The Attempt at a Solution y=sum from 0 to inf Cnxn x^2*y'=sum from 1 to inf nC n xn+1 = sum from 2 to inf (n-1) C n-1 xn = sum from 1 to inf (n-1) C n-1 xn...
  23. C

    How can I solve a separable differential equation for an initial value problem?

    Homework Statement Solve the initial value problem: dx/dt = x(2-x) x(0) = 1 Homework Equations Problem statement. The Attempt at a Solution Based on the format, I attempted to solve the problem as a separable differential equation: ∫dx/(x[2-x]) = ∫dt Evaluating to...
  24. J

    How to create a DiffEq to find mortgage prepayment savings

    I'm trying to create and then solve A D.E. that deals with mortgage amortization and mortgage prepayments. By playing around with the prepayment option on mortgage amortization calculator at http://mortgage-x.com/calculators/extra_payment_calculator.asp , I found that the ratio of money saved at...
  25. T

    A Refresher from Calc I to DiffEQ?

    I need some help, guys. I've taken Calc I -> DiffEQ, but I don't remember everything well. My NukeE courses are growing increasingly difficult (using calculus/diffeq that I forgot how to use). What would you guys recommend doing as a refresher to pick it all back up?! Thank you!
  26. M

    Frobeniuns Method/Generalized Power Series to DiffEQ solutions

    (Working out of Boas chapter 12, section 11) 3xy'' + (3x + 1)y' + y = 0 I'm asked to solve the differential equation using the method of Frobenius but I'm finding the way Boas introduces/explains/exemplifies the method to be incredibly confusing. So, I used some google-fu and was even...
  27. I

    Calc III/ Diffeq Without Calc II

    Hey Everyone! So basically my question is if it's possible to take calc III or Diffeq without having taken Calc II. I know a decent amount of Calc II but I'm not 100% sure of everything that it covers. If it would be possible, which class would be easier? I know that it definitely depends on...
  28. N

    Took DiffEq Over 5 Weeks and Didn't Learn Anything; Retake?

    Two summers ago back when I was a lazy and all-around poor student, I took DiffEq at a community college in 5 weeks because I had heard it was easier than during the regular semester. It was easy, but I didn't end up learning anything (mostly my own stupid fault), and now I'm wondering if I...
  29. A

    Basic question about separable diffeq methodology

    It's my understanding that the definition of the indefinite integral is: ∫f(x)dx = F(x) + C, where d/dx [F(x) + C] = f(x) and C is an arbitrary constant And while dx has meaning apart from the indefinite integral sign the indefinite integral sign has no meaning apart from dx. Adding an...
  30. C

    Linear first-order diffeq system for radioactive decay chain

    Homework Statement Given the followin[Sg decay chain- X→Y→Z Solve for Nx(t), Ny(t), Nz(t) for the case of Rx(t)=\alphat and assuming Ny(t)=Nz(t)=0 Homework Equations dNx(t)/dt = -\lambdaxNx(t) + Rx(t) dNy(t)/dt = -\lambdayNy(t) +\lambdaxNx(t) dNz(t)/dt = -\lambdazNz(t) +\lambdayNy(t) The...
  31. P

    Understanding and Solving DiffEq: x' = 2*x^(1/2)

    Homework Statement I have the differential equation: x' = 2*x1/2 The Attempt at a Solution I can see easily that the solution x = t2 satisfies the equation, however wolfram tells me the solution is: 1/4 (4 t2 + 4 t C + C2) which also satisfies the solution... I'm wondering...
  32. N

    Is There an Analytic Solution for This Non-Homogeneous Differential Equation?

    Is there an analytic solution for: y"(c+dx+ex^2) + ay + b = 0, y (x=0) = Ts y'(x=L) = 0 where a,b,c,d,e are all constants?
  33. V

    What is the complementary equation for this diffeq

    Homework Statement y" + 6y' + 9y = 1+x Homework Equations The Attempt at a Solution r^2 + 6r + 9 = 0 (r +3)^2 = 0 r = -3 I thought the yc = c1e-3x + c2e-3x buy my prof says yc = c1e-3x + c2xe-3x why is the x in there?
  34. V

    Where is it appropriate to place absolute value bars while solving this diffeq

    Homework Statement just a simple problem as an example dy/dx = y/x dy/y = dx/x Homework Equations The Attempt at a Solution dy/y = dx/x ln|y| = ln|x| + c then when I take exponents do I still have to include the absolute value bar |y| = |x|c
  35. L

    Is There More Than One Method to Solve Nonhomogeneous Differential Equations?

    Homework Statement y-(sinx)2)dx + sinx)dy Homework Equations Since the result wouldn't be a line, the equation would only be linear in one of its variables. The Attempt at a Solution y-(sinx)2 = 0 ; sinx = 0 --- y = (sinx)2 + sinx No clue... Also, is there more than one way to...
  36. G

    Trying to apply my DiffEq solution.

    Homework Statement S is the balance of a savings account W is the amount withdrawn per year. k is a rate percentage of continuous interest per year 1. Solve the differential Equation above. 2. Draw a phase portrait and assess the solution's stability. 3. Assume you have $1,000,000 for...
  37. B

    More reduction of order DiffEq goodness

    Homework Statement Use the substitution y = (x^2 + 1)u to solve the differential equation (x^2 +1)y\prime\prime = 2y The Attempt at a Solution I was having some trouble with these earlier because I needed to brush up on my trigonometric substitution. Let's try this one... Making the...
  38. J

    Upper level physics classes, weak DiffEq

    Is it possible to do well in upper level physics classes with a weak foundation in differential equations? I've never taken a DiffEq class. Some of the extreme basics were covered in Calc 2, and there will be a crash course in DiffEq in my Mathematical Methods in Physics class (using the Boas...
  39. jinksys

    Linear Dependence of Functions with Absolute Value

    Homework Statement Determine whether the pairs of functions are linear dependent or linearly independent. f(x) = x^3 g(x) = x2|x| Homework Equations The Attempt at a Solution g(x)=x^2|x| = x^2*sqrt[x^2] = sqrt[x^6] = x^3 f'=3x^2 g'=3x^2 fg'-f'g = 0 Linearly Dependent...
  40. F

    Classifying DiffEq: 3x+1=4t, Identify Homogeneous Parts

    I need to classify a bunch of differential equations and this one has me stuck... 3x+1=4t Would this be zeroth order? Or should I just call it a quadratic equation? Also, I need to identify the homogeneous parts of these equations. I know what a homegeneous differential equation is, but how...
  41. K

    DiffEQ: Determining the largest interval of a solution

    The question is to determine the solution to the following 1st order linear DE, along with the largest interval the solution is valid on: cosx \frac{dy}{dx} + (sinx)y=1 Rewriting it shows it to be linear: \frac{dy}{dx} + (tanx)y = secx The intergrating factor is: e^{\int{tanx...
  42. J

    DiffEq question (solving damped wave equation)

    How do you use separation of variables to solve the damped wave equation y_tt + 2y_t = y_xx where y(0,t) = y(pi,t) = 0 y(x,0) = f(x) y_t (x,0) = 0 --- These are partial derivatives where y = X(x)T(t) So rewriting the equation I get X(x)T''(t) + 2X(x)T'(t) = X''(x)T(t) which...
  43. A

    Integration within a DiffEQ problem

    Homework Statement Solve the given differential equation by using an appropriate substitution. Homework Equations (x^{2}+xy+3y^{2})dx-(x^{2}+2xy)dy=0 y=ux, dy=udx+xdu The Attempt at a Solution (x^{2}dx+ux^{2}dx+3u^{2}x^{2}dx)-(ux^{2}dx+x^{3}du+2u^{2}x^{2}dx+2ux^{3}du)=0 After...
  44. J

    Maximizing Mortgage Borrowing Potential with Continuous Interest and Payments

    I am having trouble understanding conceptually the following DiffEq problem: "Suppose you can afford no more than $500 per month of payment on a mortgage. The interest rate is 8% and the mortgage term is 20 yrs. If the interest is compounded continuously and payments are made continuously...
  45. M

    How Do I Normalize a Quantum Mechanics Equation With an Integral?

    I am starting to teach myself quantum mech. in preperation for this coming semester, however i have hit a mathamatical road block. I need to normalize the folowing equation. P(x)=Ae^{-\lambda(x-a)^{2}} Unfortunatly I do not take DiffEQ until this coming semester and I don't know how to...
  46. K

    How would I solve a DiffEq of the form

    \frac{dn(t)}{dt} = A sin(B*n(t)*t) n(t) Or a more general \frac{dn(t)}{dt} = F(n(t)) n(t) I'm not even sure what method I could use, or what it would be called. A first order, non-linear equation? Maybe it looks neater as: \frac{dn}{dt}=A n Sin(n t) EDIT : This isn't homework. I'm just...
  47. M

    Solve 4th Order Differential Equations (No Guesswork!)

    [SOLVED] Forth Order DiffEq I've recently come across the following differential equations. y''''+y=0 and y''''-y=0. Can differential equations such as these be solved with any technique other than guessing for the particular solutions? They seem very simular to trig's equation but are still...
  48. T

    AP Mechanics DiffEQ: Restoring Torque & Angular Acceleration

    Homework Statement A stick of length 2L and negligible mass has a point mass m affixed to each end. The stick is arranged so that it pivots in a horizontal plane about a frictionless vertical axis through its center. A spring of force constant k is connected to one of the masses. The system...
  49. T

    DiffEq - Initial Value Problem / Integration help

    This question is an initial value problem for diffeq. We are asked to solve explicitly for y. (1+cos(x))dy = ((e^(-y))+1)*sin(x)dx , y(0) = 0 I attempted a separation of variables and ended up with the following: dy / ((e^(-y))+1) = (sin(x) / (1 + cos(x))) dx I know that my...
  50. 0

    DiffEq vs. Biology: Which to Choose?

    Is introductory differential equations boring? I can take another course and DiffEq is what I am considering mainly because it is mentioned a lot. I have no real need to take this course, unless it is useful in AI. If I take the course it will be taught at a general level in a class of 60...