Differential equations Definition and 999 Threads

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. R

    Transient heat transfer in a sphere

    Homework Statement A metal sphere is subjected to a heat flux, 5000 W/m2. It is originally at 20 C. How long does it take to heat to 90 C? D = 5 cm density = 8522 kg/m3 cp = 0.385 kJ/kg-K k = 104 W/m-k Homework Equations rate of heat input = rate of heat accumulation -k*A*dT/dr = m*cp*dT/dt...
  2. faradayscat

    Differential equation with power series

    Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
  3. Dusty912

    Match second-order differential equations with their y(t) graph

    Homework Statement The problem is in the picture. #17 I would have typed it but there are graphs that are needed for solving it. Basically trying to figure out which y(t) graph belongs to one of the sinusoidal forced equations with various parameters. Homework Equations euler's formula...
  4. D

    System of Differential Equations

    Homework Statement dx/dt + dy/dt = 2x + 2y + 1 dx/dt + 2(dy/dt) = y + 3 Homework Equations The Attempt at a Solution Dx + Dy = 2x + 2y + 1 Dx + 2Dy = y + 3 --Rearranging---------------------------- Dx - 2x = -Dy + 2y + 1 Dx = -2Dy + y + 3 ---Factoring----------------------------- (D...
  5. E

    A Nonhomogeneous second order nonlinear differential equations

    Hello every one, I have an equation related to my research. I wonder if anyone has any suggestion about solving it? y''+y' f(y)+g(y)=h(x) thanks
  6. kostoglotov

    Why are these capacitor voltages opposite polarity?

    Homework Statement imgur link: http://i.imgur.com/JETj0Cq.png Homework Equations V_L = L\frac{dI_1}{dt} V_L + V_1 + V_2 + V_{R1} = 0 The Attempt at a Solution [/B] Just using basic KVL, shouldn't V_L = -V_2 - V_1 - V_{R1} and so therefore we should get \frac{dI_1}{dt} = \frac{-V_2 -...
  7. The Bill

    Analysis General texts on systems of partial differential equations?

    What are some good general textbooks on the properties and solution of systems of partial differential equations? I'm most interested in the general theory of vector and tensor valued PDEs like Maxwell's, Navier-Stokes, and the bulk equations governing elasticity and deformation of solids, etc...
  8. Poetria

    Conte Riccati and Jakob Hermann

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  9. E

    I Question about using matrices for differential equations

    Let x(t)= [x1(t) x2(t)] be a solution to the system of differential equations: x′1(t)=−2x1(t)+2x2(t) x′2(t)==−6x1(t)+9x2(t) If x(0)= [4 -2] find x(t). I got the eigenvalues to be -6 and -5, but I don't know how to calculate the coefficients in front of the exponents. For lambda=-6 I get...
  10. S

    I Question about solution to Laplacian in Spherical Polars

    I was following this derivation of the solution to the Laplacian in spherical polars. I was wondering where the two equations ##\lambda_{1} + \lambda_{2} = -1## and ##\lambda_{1}\lambda_{2} = -\lambda## come from? Thanks.
  11. The-Mad-Lisper

    Need Help with Integration for Solving ODE

    Homework Statement \frac{dy}{dx}=y^2-1 y(0)=3 Homework Equations \frac{dy}{dx}=f(y) \leftrightarrow \frac{dx}{dy}=\frac{1}{f(y)} The Attempt at a Solution \frac{dx}{dy}=\frac{1}{y^2-1} dx=\frac{dy}{y^2-1} \int dx=\int \frac{dy}{y^2-1}+C x=\int \frac{dy}{y^2-1}+C How do I integrate \int...
  12. Dusty912

    Differential Equations: linearity principle

    Homework Statement Consider the linear system: dx/dt=x-y dy/dt=x+3y a. show that the function (x(t), y(t))=(te2t, -(t+1)e2t) is a solution to the differential equation (easy) b. Solve the initial value problem dx/dt=x-y dy/dt=x+3y y(0)=(0,2) need help with part b not a Homework Equations...
  13. Elvis 123456789

    How to represent this complex number?

    Homework Statement Finding "polar" and "rectangular" representation of a complex number? Make a table with three columns. Each row will contain three representations of a complex number z: the “rectangular” expression z = a + bi (with a and b real); the “polar” expression |z|, Arg(z); and a...
  14. Elvis 123456789

    How to go from a*cosx + b*sinx TO A*cos(wt - phi)?

    Moved from non-homework forum section, so homework template is not present. Express Cos(t - pi/8) + Sin(t - pi/8) in the form A*Cos(wt - phi). I got sqrt(2)*Cos(t-3pi/4). Not sure if that's right though
  15. G

    MHB System of differential equations involving matrices

    Let $\mathbf{A} = \begin{pmatrix} 9 & 2 \\ 1 & 8 \end{pmatrix}.$ Obtain the general solution $\mathbf{y}(t)$ of the system of differential equations $\displaystyle \frac{d\mathbf{y}}{dt} = \mathbf{Ay}$: $\begin{cases} \frac{dy_1}{dt} = 9y_1+2y_2 \\ \frac{dy_2}{dt} = y_1+8y_2 \end{cases}$...
  16. chwala

    A problem on differential equations

    Homework Statement A boy is eating a 250 gram burger. When he has eaten a mass m grams, his rate of consumption is given by ## 100-1/900m^2## grams per minute. How long does it take him to finish the meal?[/B]Homework EquationsThe Attempt at a Solution ##dm/dt=100-1/900m^2 ...
  17. L

    (Tricky) Comet Picking up Mass - Differential Equations

    Homework Statement A comet in deep space picks up mass as it travels through a large stationary dust cloud. It is subject to a gravitational force of magnitude Mf acting in the direction of its motion. When it entered the cloud, the comet had mass M and speed V. After a time t, it has traveled...
  18. faradayscat

    Differential equations particular solution

    Homework Statement Particular solution of y" - y' - 2y = e^(2x) Homework Equations None The Attempt at a Solution This makes no sense to me, why do I have to use the solution of the form y(t) = cxe^(2x) For the problem above, but when I switch the signs and it becomes y" - y' + 2y =...
  19. B

    A Differential Equations not solvable

    In the first image it shows the ##\alpha^2-w_0^2<0## situation whereas in the second image the situation is when ##\alpha^2-w_0^2=0##.The problem is the book says to use ##T_0=2\pi/w_0## to determine diabetes but you can't do that when ##\alpha^2-w_0^2=0## because it can't be put into a cosine...
  20. W

    Help with nondimensionalization

    I am working on an assignment for my neuroscience course, and I'm running into a problem with one question which requires me to rewrite an equation into its nondimensionalized form. The equation is given below. and I need to convert it to the form by rescaling and shifting the given...
  21. E

    (Differential Equations) Determining Linearity of a Function

    This is more of a general question than a specific homework question, because it popped up in more than 1 problem. If you have 'x' has the independent variable and 'y' as the dependent variable, you can determine the linearity in 'y' by seeing if any of the derivatives (dy/dx) are being raised...
  22. kostoglotov

    System of ODEs in a rotating coord. system

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  23. L

    Basic question about certain second order PDE's

    Given is the second order equation, ##X_{uv} = A(u,v)X_{u} + B(u,v)X_{v}## defined on a domain ##(u,v)## in the plane. ##X## is a three dimensional vector and ##A## and ##B## are arbitrary smooth functions. When does such an equation determine a surface in ##R^3## and what in general can be...
  24. H

    Classifying second-order Partial differential equations

    What does it mean when it says to classify the second-order partial differential? (See attached) How would I get started?
  25. B

    Wronskian and differential equations

    Homework Statement The problems are in the uploaded file. 18) satisfies the differential equation y''+p(t)*y'+q(t)*y=0 p(t) and q(t) are continuous Homework Equations Wronskian of y1 and y2 The Attempt at a Solution 18) I don't really get this one 19) Solved most but at the end, where I...
  26. D

    Coupled Differential Equations

    Homework Statement Hi. I am trying to solve a problem on renormalisation group flow and have come across the following coupled equations that I need to solve: Λ ∂g/∂Λ = b.m Λ ∂m/∂Λ = -2.m + a.g Where a and b are just constants. I need to find g(Λ) and m(Λ). Homework Equations [/B] I...
  27. faradayscat

    Brainstorm Ideas for Project - Get Suggestions on Differential Equations

    I will be writing a paper by modelling a system and its differential equations. I have just started brainstorming ideas but I would also like to hear other peoples' ideas as well! I have these ideas so far, can anybody suggest relevant articles or papers on these topics (avoid wikipedia if...
  28. P

    Why differential equations? When to use them?

    Hello everyone, I have some questions that have been bothering me for years now. Why do we use differential equations? When would I normally think about using one? What would make me use one? I know it means the slope of a function and I know the derivation/integration rules + the solution of...
  29. G

    Why 2nd order differential equation has two solutions

    Hello. I forgot the reason why 2nd order differential equation has two independent solutions. (Here, source term is zero) Why 3 or 4 independent solutions are not possible? Please give me clear answer.
  30. B

    Unpacking the Implication of a Formula: n=2

    Edited by moderator In the attached file it says that the formula for when n=2 implies the next one but I don't understand why it suddenly became (s-t0)^2 inside the integral instead of (s-t0) in the next one and keeps this pattern, it doesn't seem to be implied anywhere for me.
  31. Dean Whaley

    Solve for a constant in an equation

    Im trying to solve for a constant in an equation and it involves taking the arctanh(6.55) and my calculator is giving me an error, is there a way around this?
  32. I

    What are differential equations good for?

    When I learned differential calculus, my instructor spoke only in terms of x's & y's. I could differentiate, but had absolutely no idea what I was doing or how it applied. My mentor pulled me aside, gave me a function, f(x)=x^2, & made note that the function was modeling a vehicle's position in...
  33. J

    What kind of differential equations one must know for QM?

    I will be taking a first course on Quantum Mechanics and just wanted to know what kind of ordinary differential equations must i know before going into the course. Thank you!
  34. C

    How do differential equations relate to traffic flow?

    Hello, I'm trying to learn about the role that differential equations play in traffic flow, and how I can use them to model/predict/whatever you do with them with traffic flow. Do you guys know of any good and in-depth (preferably online and free) resources I can use to learn about this?
  35. U

    Good source homework problems for beam spring structures?

    Hello there, In this semester I'm studying the vibrations course. In this course we get questions like calculating the total stiffness, set up differential equations and so on. The book that we use (Theory of Vibrations) has not been very useful, our exams give beam spring structures related...
  36. A

    Solving Second Order Linear Diff Eqns: Proving Theorems & More

    So, I had studied oscillatory motion for a while and I found it unpleasant to have to remember the various different solutions for the equations of motion. I began to learn about second order linear differential equations and now I know how to solve this kind of stuff. But there is a problem...
  37. noowutah

    Specify function given certain constraints

    Let F:V\rightarrow{}\mathbb{R}^{+}_{0} be a differentiable function. V is the set of all positive real-valued 2\times{}2 matrices, so V=\left\{\left[ \begin{array}{cc} a & b \\ c & d \\ \end{array}\right]\mbox{ with }a,b,c,d\in\mathbb{R}^{+}\right\} Here are the two constraints for F...
  38. J

    Applied Any books on Ordinary Differential Equations w/ applications

    Hello, I really need a good book on ordinary differential equations with applications on Quantum Mechanics, as I will be attending a course on QM but I do not have the proper mathematical background that is needed.
  39. B

    Setting up the differential equations on a moving slope.

    We have a slope, which has a M mass and a particle with m mass. There are no friction forces between the slope and the ground nor with the particle. The question is, how can i set up the differential equations to get t time. The slope has an h height and an angle of alpha with the ground...
  40. N

    Finding critical points for phase planes

    Homework Statement X'=x-y^2 Y'=x^2 -xy -2xHomework Equations Find the critical point of this system. The Attempt at a Solution I know one point is (0,0). When I tried it I got sqrt(x)= 2 and sqrt(x)= 1 but the point (1,1) doesn't fit later parts of the question. I'm also aware that since a...
  41. H Smith 94

    Finding a solution to Laplace's equation

    So here I have Laplace's equation with non-homogeneous, mixed boundary conditions in both x and y. 1. Homework Statement Solve Laplace's equation \begin{equation}\label{eq:Laplace}\nabla^2\phi(x,y)=0\end{equation} for the following boundary conditions: \phi(0, y)=2; \phi(1, y)=0; \phi(x...
  42. N

    Linear algebra and differential equations advising

    Hey guys, I'm talking to my advisor soon and I was wondering if it is typical to, after taking multi variable calc, to take differential equations and linear algebra simultaneously? I'm going to have to be taking both modern physics and organic chemistry II as well, for context. Thanks everyone
  43. L

    Ordinary differential equations. Series method.

    Question: Why equations x(1-x)\frac{d^2y}{dx^2}+[\gamma-(\alpha+\beta+1)x]\frac{dy}{dx}-\alpha \beta y(x)=0 should be solved by choosing ##y(x)=\sum^{\infty}_{m=0}a_mx^{m+k}## and not ##y(x)=\sum^{\infty}_{m=0}a_mx^{m}##? How to know when we need to choose one of the forms. Also when I sum over...
  44. S

    Calculus Good book on differential equations

    I need a good book on differential equations.
  45. M

    MHB System of differential equations

    Hey! :o When we have a system of differential equations $$\bigwedge_{j=1}^n L_j y=f_j$$ is it possible that we reduce it into one differential equation $Ly=f$ so that the system has a solution iff the differential equation $Ly=f$ has a solution? (Wondering)
  46. C

    MHB Stochastic differential equations

    Hello to all, I am a new member, but I've been reading and getting help from this forum for a year! I recently started to study about stochastic calculus because I am considering risk management/ actuarial/ finance job. I would appreciate your help. If we have Poisson $(\lambda)$ and $W$...
  47. P

    Differential Equation with Bessel Function

    <<Moderator note: Missing template due to move from other forum.>> Good afternoon. I'm trying to solve a differential equation with bessel function solutions. I am trying to solve \begin{equation*} y''(x)+e^{2x}y(x)=0 \end{equation*} using the substitution ##z=e^x##. The textbook this problem...
  48. N

    Using Abel's Theorem, find the Wronskian

    Using Abel's thrm, find the wronskian between 2 soltions of the second order, linear ODE: x''+1/sqrt(t^3)x'+t^2x=0 t>0 I think I got the interal of 1/sqrt(t^3) to be 2t/sqrt(t^3) but this is very different to the other examples I've done where a ln is formed to cancel out the e in the formula...
  49. MidgetDwarf

    When Can I learn Partial Differential Equations?

    Is my background enough to learn partial differential equations? I have completed up to calculus 2 and linear algebra. I am currently taking Cal 3 and Ordinary Differential Equations. I am doing well in both courses. I would like to learn PDE and a bit more Linear Algebra, during the winter...
  50. C

    Newtonian gravity and differential equations

    Hello, So i know that the gravitational acceleration experienced by a body is -GM/||d||^2 * dhat, where dhat is the current displacement unit vector, which has a magnitude of 1. the magnitude of a vector is equal to to the square root of the sum of its squared components. This will be a 2d...
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