What is Differential equations: Definition and 999 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. D

    Writing Linear Differential Equations as Matrix Differential equations

    Homework Statement Be able to write a system of Linear homogenous differential equations as a matrix differential equations. Homework Equations The Attempt at a Solution I have uploaded the work and original problem. My question is did I properly answer this question?
  2. D

    3x3 Matrix Differential Equations

    Homework Statement Find the general solution to the system of differential equations. Homework Equations The Attempt at a Solution I uploaded the original equation and my work so see the attachment. I want to know how they got the vectors the got typically when I have done 2x2...
  3. F

    Second order differential equations

    Homework Statement finding the general solution. I would like to know if the 20-e^x will be treated as a sum of a polynomial? Homework Equations The Attempt at a Solution
  4. C

    Basic Differential Equations Problem (I Have No Idea What I Am Doing)

    Homework Statement Denote by L(t) the length of a fish at time t, and assume that the fish grows according to: dL/dt = k(34-L(t)) with L(0) = 2 a) solve the above equation b)Use your solution in part a to determine k under the assumption that L(4) = 10. c) Find the length of the fish...
  5. D

    Systems of Homogeneous Linear Differential Equations

    Homework Statement I uploaded the problem statement please see attachment for original problem. The problem number is 4. Homework Equations The Attempt at a Solution For clarity I uploaded what I have done please see the attachment with my work on it. I am not sure if I am doing...
  6. T

    Help How to solve a set of 12 partial differential equations with 8 variables.

    Hi, I am a master student comes from USM in Malaysia. My main focus is on monopole instanton solution in static form which did not include time, i using the Lagrangian to generate out the 12 set equations of motion with 8 variables, the software that i use is Matlab with Optimization toolbox-...
  7. K

    Origin as the Only Critical Point: Solving Differential Equations

    Homework Statement Show the origin is the only critical point Homework Equations x'= -x-x3 y-= -y-y5 The Attempt at a Solution I'm not really sure how to go about this. I missed a few lectures due to a medical issue, and now were at the end of the semester and I can't get in touch with the...
  8. N

    Differential Equations and Circuits

    So, I'm learning how to solve LR, RC, LC etc. types of circuits using differential equations. I understand how to do the math with differential equations, but I am confused as to why the variables are split in the way they are. For example, for an LR circuit you have the equation...
  9. D

    What to review from linear algebra for a differential equations class?

    I'm going to be taking a class in ordinary differential equations over the summer and have about 2 weeks to prepare. The class has linear algebra as a prerequisite, and I just wanted to know what I would need to review from linear algebra to prepare myself for the course?
  10. S

    Differential Equations - Eigenvalues and Eigenfunctions

    Homework Statement Find the eigenvalues and the eigenfunctions for x^2y"+2xy'+λy = 0 y(1) = 0, y(e^2) = 0 Homework Equations See problem The Attempt at a Solution My book has one paragraph on this that does not help me. I tried using an auxiliary equation and solving for lambda...
  11. Q

    MATLAB Solve matrix differential equations using Matlab

    For d V/d t=AV + VA^{T}+ D, where A and D are the given 6*6 matrices, A=[-0.5*K 0 0 0 0 G1;0 -0.5*K 0 0 -G1 0;0 0 -0.5*K 0 0 -G2;0 0 0 -0.5*K -G2 0;0 G1 0 -G2 -0.5*RM 0;-G1 0 -G2 0 0 -0.5*RM] D=[0.5*K 0 0 0 0 0;0 0.5*K 0 0 0 0;0 0 0.5*K 0 0 0;0 0 0 0.5*K 0 0;0 0 0 0 0 0;0 0 0 0 0...
  12. S

    Help with interpretation of question. (Differential equations)

    http://dl.dropbox.com/u/33103477/Linear%20oscilator.png I am having trouble understand the question, what I have done its solve the equation using the substitution x=e^{rx} Then, I have the solution given by: x(t)=c_1 e^{t(\sqrt{\gamma^2 - \omega^2 })} + c_2e^{-t(\sqrt{\gamma^2 -...
  13. J

    Mathematica Help in Mathematica code for solutions expansion of differential equations

    I need to NDsolve four differential equations. But I find my code does not work. Who can give me some suggestions? Thanks!
  14. N

    A question about differential equations

    Hi,I'm new to these and thus my question might sound stupid: Do differential equations ALWAYS have just one or zero general solutions? I know each diff.equation can have multiple particular solutions, but can it only have one or zero general solutions?
  15. T

    Mechanical Motion of Springs Differential Equations

    Homework Statement The Attempt at a Solution So I've been interpreting the information in the problem as follows: F_{damping} = 4u' = μ(u'), k = \frac{4N}{m}. If the system is critically damped then μ = 2\sqrt{km} = 2\sqrt{\frac{4N}{m}m} = 2\sqrt{4N}. Now it seems as though the spring...
  16. D

    General solutions of linear differential equations, why add the homogen. + particular

    I've come across an issue that was bugging me last semester in my circuits class today: Finding general solutions of linear differential equations. Just add the homogeneous and particular solution and it's done. Last semester it wasn't explained why exactly this is possible and this semester...
  17. T

    Solving Matrix of Differential Equations With Initial Values

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  18. B

    First order differential equations.

    Hi, I have problems rewriting equations with the term y'' as a system of first order differential equations. I've been given several equations and was told to write them as 1st order DEs, then calculate the numerical solution using the Euler's modified method. I know that y'=f(x,y), so if...
  19. J

    Second Order Differential Equations

    Homework Statement Solve the ODE with the boundary conditions given Q''+Q = Sin(2x) where Q(0) = 1 and Q'(0) = 2 So i know i need to solve the general and particular solutions, however, I am a little confused. Any help or advice would be great, Thanks in advance. Homework Equations Y...
  20. D

    Differential Equations question.

    Homework Statement We have this DE: y(t)''+2y(t)'+y(t)=e^(-t) y(0)=1 y(0)'=0 Homework Equations We use LT L(y(t)'')=F(s)s^2-y(0)s-y(0)' L(y(t)')=F(s)s-y(0) L(y(t))=F(s) The Attempt at a Solution After calculus F(s)=(s^2+3s+3)/(s+1)^3=1/(s+1)+1/(s+1)^2+1/(s+1)^3 => y(t)=e^(-t)(1+t+t^2/2) Which...
  21. A

    MATLAB MATLAB: How to solve a system of Nonlinear Differential Equations?

    Hello to everybody, I'm very new to solving ODES and equations with MATLAB. I have been asked to solve a system of nonlinear equations for simulating the growth of a solid tumor. Assuming that we have the 5 unknowns which are dxd arrays: f,g,m,p and n. f(x,t) is the volume fraction of tumor...
  22. T

    Confused about system of linear differential equations?

    Homework Statement So in my notes it says that the general solution to a system of linear equations: x1'(t) = a11x1(t) + a12x2(t) x2'(t) = a21x1(t) + a22x2(t) is: x(t) = c1eλ1tv1 + c2eλ2tv2 What I don't understand is how you go from the system of equations to the general solution, the notes...
  23. T

    Modelling with differential equations

    Homework Statement The Attempt at a Solution I think that problems such as this one tend to take on the rough form of \frac{dQ}{dt} = rate in - rate out. I suppose I should treat each lake such that is has it's own equation regarding concentration. I reasoned that, in the case of the...
  24. O

    Solving 2 differential equations

    dx/dt=ay and dy/dt=bx where x and y are function of t [x(t) and y(t)] and a and b are constant. 1) show what x and y satisfy the equation for a hyperbola: y^2-(b/a)*x^2=(y_0)^2-(b/a)*(x_0)^2 2) suppose at some time t_s, the point (x(t_s),y(t_s)) lies on the upper branch of hyperbola, show...
  25. T

    Linearity in Differential Equations

    Homework Statement Is the following differential equation linear: yy' + 2 = 0 The Attempt at a Solution I have the definition of linear as being a_0 (t) y^{(n)} + a_1(t) y^{n-1} + a_2 (t) y^{n-2} ... = 0. Now, presumably y is a function of t. Thus, I could define y = a_0 (t) and...
  26. T

    Differential Equations Water Evaporation

    Homework Statement A pond contains 1,000,000 gal of water and an unknown amount of chemical. Water containing .01 gram of this chemical per gallon flows into the pond at a rate of 300 gal/hour. The mixture flows out at the same rate, so the amount of water in the pond remains constant. Assume...
  27. C

    Banach Fixed Point and Differential Equations

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  28. A

    Differential equations to determine water level

    Hi, the problem i have is this: How long will it take a water reservoir with an Average level of 300,000,000 m3 to drop to 90% of the average level, if there is a drought, taking into account average rainfall, evaporation and amount of water taken in and out? Assumption are: Reservoir is...
  29. T

    Laplace transforms of differential equations

    Homework Statement I've attached the problem.Homework Equations L(1) = 1 /s L(t^n) = n!/s^(n+1)The Attempt at a Solution because the question only asks for X(s) I only considered the x' + x + 4y=3 equation. I applied laplace tranforms and got: s X(s) - x(0) + 1/s^2 + 4/s^2 = 3/s. since x(0) =...
  30. C

    Science & engineering math: system of differential equations

    Homework Statement Solve the system of differential equations: y'(t) + z(t) = t y"(t) - z(t) = e-t Subject to y(0) = 3, y'(0) = -2, and z(0) = 0 Homework Equations My professor did an example in class that was much simpler and solved it using Kramer's rule. The Attempt at a...
  31. J

    About simple differential equations

    when i solve dy/dt= y-b (1/y-b)(dy/dt)=1 d(ln│y-b│)/dt=1 when i integrate both sides respect to t, ln│y-b│=t+c (c is a constant) y=±e^(at+c)+b =±c1*e^at + b (c1 is a constant) then the book replaces ±c1 with c2 (constant) but isn't it wrong to do so? Because c2 can't show that...
  32. S

    Differential Equations or Number Theory for Computer Science?

    I'm getting ready to register for classes for the fall. To make a long story short, I might have to take another math class to satisfy a degree requirement, rather than a computer science class. I'm taking Linear Algebra right now. I enjoy it, and it seems to have a lot of practical...
  33. T

    Bernoulli Differential Equations

    Suppose I have a Bernoulli differential equation; that is, an equation of the form: y' + p(x)y = g(x) y^n. Supposing that I let n=1, the equation is linear. Can I solve it by constructing an integrating factor? That is, can I observe: y' + p(x)y = g(x) y → y' + y[(p + g)(x)] = 0. I would then...
  34. R

    Differential Equations Question

    Homework Statement http://img694.imageshack.us/img694/6672/37517439.jpg The Attempt at a Solution For part (a), according to the uniqueness theorem, if f(t,y) and ∂f/∂y are continious in a given interval in which (t0, y0) exists, and if y1(t) and y2(t) are two functions that solve the...
  35. A

    Proving the multiverse theory through partial differential equations

    Can this be done? If so, how can I go about doing so? This is merely a potential science fair idea for 2013.
  36. J

    Help with Differential Equations please

    Homework Statement \frac{dy}{dx}=3y f(2)=-1 and \frac{dy}{dx}=e^{y}x when x=-2 y=-ln(3) I'm in Calc AB By the way, so please do not try to show me methods that are too advanced. Homework Equations There are no relevant equations? The Attempt at a Solution My attempt at the first...
  37. S

    Higher Order Differential Equations: Variation of parameter.

    Hi, I'm not exactly sure how to solve the following non-homogeneous ODE by variation of parameters. Solve the given non-homogeneous ODE by the variation of parameters: x^2y" + xy' -1/4y = 3/x + 3x Can someone please point me in the right direction? Help will be much appreciated...
  38. N

    Differential Equations - Logistic Model

    I have the equation dP/dt = kP(1 - P/A). It is supposed to describe a logistical situatuon involving the carrying capacity of the system. k is a constant, and A is the carrying capacity of the system. t is time and P is population as a function of time. P(0) = P0. I solved c (the integration...
  39. A

    Differential equations and physics proof problem

    Homework Statement The kinetic energy K of an object of mass m and velocity v is given by K=1/2mv2. Suppose an object of mass m, moving in a straight line, is acted upon by force F=F(s) that depends on position s. According to Newton's Second Law F(s)=ma. A is acceleration of the object...
  40. P

    Differential Equations - Reduction of order

    General Question. How would I recognize when to do a full reduction of order versus using the formula y2=y1 integral(e^integral(P(x)dx)/y1^2) So far I only know by when the homework problem set, specifies to use the formula or the reduction of order. However I want to know if I can tell...
  41. P

    2nd order differential equations

    y''-3y'+2y=e^t y(0)=0 y'(0)=-1 yh=solution to homogeneous equation (y''-3y'+2y=0) = Ce^t+Ae^(2t) C and A are constants yp=solution to particular solution (e^t) yp=ae^t where a is a constant. It turns out that this solves the homogenuous solution so I had to multiply it by a...
  42. A

    Why am I struggling with Differential Equations?

    Why am I struggling with Differential Equations?? Please help: I did well in Calc I-III, and now am struggling in Diff.Eq. Anyone else find themselves in the same situation, and how did you save yourself? TIA:)
  43. P

    Differential equations - interval of existence

    dy/dx=(sinx)/y Initial condition is y(pi/2)=1 The solution to the IVP is y=(1-2cosx)^.5 That I know is correct, but they're saying the interval of existence is when pi/3<x<5*pi/3. Is that wrong? I think it should include the π/3 and 5π/3.
  44. J

    Differential Equations question

    Homework Statement Determine for which values of m the function ∅(x) = xm is a solution to the given equation a) 3x2y" + 11xy' -3y = 0 b) x2 y" - xy' - 5y = 0The Attempt at a Solution I tried approaching this problem by substituting ∅(x) into the question. a) 3x2(xm)'' + 11x(xm)' - 3(xm) = 0...
  45. mnb96

    System of two differential equations with trigonometric functions

    Hello, do you have any strategy to suggest in order to solve the following system of partial differential equations in x(s,t) and y(s,t)? \frac{\partial x}{\partial t} = x - \frac{1}{2}\sin(2x) \frac{\partial y}{\partial t} = y \; \sin^2(x) (note that the partial differentiation is always with...
  46. I

    Solving a systems of differential equations in terms of x(t) and y(t)

    Homework Statement x' ={{-1,1},{-4, 3}}*x, with x(0) = {{1},{1}} Solve the differential equation where x = {{x(t)}, {y(t)}} Homework Equations The Attempt at a Solution I have e^t*{{1},{-2}} + e^t*{{t},{2t+1}} but I'm not sure how to get it in terms of what it's asking.Edit: Please quick...
  47. B

    Mixing Tank, Differential Equations Problem

    Homework Statement a larg tank is filled with 500 gals of water with 400lbs of salt. pure water is pumped into the tank at a rate of 3gal/min. the well mixed solution is pumped out at a rate of 7 gal/min. I need help finding the Differential Homework Equations The Attempt at a...
  48. K

    Two Partial Differential Equations

    Homework Statement This is the first problem of the two. Homework Equations The Attempt at a Solution Using separation of variables, I end up with T'(t)= -λKT(t) and X''(x)+(β/K)X'(x)/X(x)= -λ. At first I chose the negative lambda because I saw that U(0,t) and U(L,t) needed to oscillate...
  49. N

    Differential Equations - Verifying a solution of a given DE

    Homework Statement Verify that the indicated funciton is a solution of the given Differential Equation. c1 and c2 denote constants where appropriate. \frac { dX }{ dt } =(2-x)(1-x);\quad \quad \ln { \frac { 2-x }{ 1-x } } =tThe Attempt at a Solution I'm not quite sure how to really start...
  50. M

    Differential Equations Method of Flexible guessing

    Homework Statement Find a particular solution Yp of the given equation. Primes denote deriviate with respect to x (method of flexible guess) Homework Equations 4y''+4y'+y=3xe^x The Attempt at a Solution when I used y=Ae^x as guess my A depended on x So y=Axe^x gave me 12A+9Ax=3x...
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