Differential equations Definition and 999 Threads

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...

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...

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...

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...

Homework Statement Find the value of x, correct to three decimal places for which: \int^{x}_{0}\frac{t^{2}}{1+t^{2}}dt=\frac{1}{2}. Homework Equations Banach's Fixed Point Theorem Picard's Theorem? The Attempt at a Solution I'm not sure where to start with this type of problem...

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) =...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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:)

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.

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...

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...

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...

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...

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...

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...

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...

Homework Statement When you turn on an electric heater, such as "burner" on a stove, its temperature increases rapidly at first, then more slowly, and finally approaches a constant high temperature. As the burner warms up, heat supplied by the electricity goes to two places. i.) Storage in...

Homework Statement A biologist prepares a culture. After 1 day of growth the biologist counts 1000 cells. After 2 days he counts 3000. Assuming a Malthusian model what is the reproduction rate and how many cells were present initially. Homework Equations P(t) = Ce^{rt} The...

Homework Statement http://img192.imageshack.us/img192/6482/asderl.jpg Find S(t). Homework Equations I managed to find these equations, but don't know how to solve them completely. mx``=Rsinφ my``=Rcosφ-mg xtgφ+y=h x`(0)=0; x(0)=0; y`(0)=0; y(0)=hThe Attempt at a Solution Need to find...

Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...

Homework Statement For the system of differential equations \frac{dx}{dt}=(-3x-y) \frac{dy}{dt}=(-2x-2y) (a) Find the general solution. (b) Find the solution if x(0)=1 and y(0)=2. Homework Equations The Attempt at a Solution I have absolutely no clue how to do this. I have never seen a...

I'm having trouble understanding what uniqueness is/means. When given a slope/direction field I don't know what I should be looking for if asked to determine if a given initial condition has a unique solution. Example: \textit{y' = }\frac{(x - 1)}{y} With this equation I can see that as long...

i'm stuck on three problems on my homework - I've been trying to solve them forever! please help? thanks in advance. PS I'm new on here and don't know what to do so I'm just going to type out the problems haha 1) Evaluate: integral of the absolute value of x^2-9 dx on interval [-4, 6] 2)...

Hi all, Back on MHF, my Differential Equations Tutorial was beneficial to many members and I was proud of my contribution to the site. However, with the fall of MHF, I have not been able to retrieve my old posts. So my plan is to start a completely new one on this site -- from scratch. I...

I am currently taking differential equations in college but I am having trouble understanding a few things. Our teacher is from Russia so it makes it a bit harder for me to understand him and sometimes for him to understand our questions... Currently I have taken Calc 1, 2, 3 and Linear...

Homework Statement Solve the differential equation dy/dx = 3x2(1+y2)3/2 Homework Equations The Attempt at a Solution So far this is what I have (I'm almost finished) - ∫dy/(1+y)3/2 = ∫3x2 dx Let y = tan(u) , dy = sec2(u) Then (1+y2)3/2 = (tan2(u)+1)3/2 = sec3(u) and u =...

Before taking Differential Equations/Linear Algebra in my first year of college, I am brushing up on Algebra I/II, Pre-Calculus, and Calculus I/II. Should I also run through Trigonometry as well or should I not waste my time on it if I don't have enough time to go through everything else first...

Homework Statement 1. For the system of equations x'(t) = 4y and y'(t) = x, obtain the equation of the trajectory (path in the phase plane) that passes through (2, 0). For this trajectory, what is the equation of the slant asymptote that (x(t), y(t)) approaches as t goes to infinity? 2...

Homework Statement Vibration in a system can be a source of problems. For example, the deck on a ship could vibrate due to the engine which represents a forcing function. This system may be simply modeled by a mass, representing the deck, a spring representing the stiffness of the deck and a...

Are you a software engineer using differential equations or more advanced math everyday to solve problems? I would like to hear about what you do--what projects do you work on? Do you consider that differential equations and advanced math skills are useful/in demand in the realm of software...

Ok guys, I've got an issue with a coupled differential equation and I just can't get to solve it: \frac{\partial r}{\partial t} = Q\frac{\partial c}{\partial t} Obviously, r depends on c and visa versa, but they both depend on time. Is there a way to uncouple these variables and solve the...

I’m reviewing differential equations after taking the course about 5-6 years ago and I have a couple of questions about the solutions of differential equations. 1) First why is the general form of the solution to linear homogenous differential equations, with non-equal and real roots to the...

can you help me ?? A 200 liter tank initially contains 100 liters of water with a salt concentration of 0.1 grams per liter. Water with a salt concentration of 0.5 grams per liter flows into the tank at a rate of 20 liters per minute. Assume that the fluid is mixed instantaneously and that...

Homework Statement dy/dt=y((3t^2)-1), y(1)=-2 Homework Equations Basic integrals The Attempt at a Solution integrate on both sides: dy/y=dt((3t^2)-1) ========>ln(y)=(t^3)-t+c ========>y=e^((t^3)-t+c) ========>y=e^((t^3)-t)e^(c) I am not sure if its some e rule that I forgot...

Hi guys, I'm currently in computer science program and I have an urgent feeling that I need better exposure to math. I have taken Discrete Math, Calculus i, ii, iii and I've independently studied linear algebra. I guess my concern is lack of differential equations and numerical methods. In...

The Wikipedia article regarding Lagrangian Mechanics mentions that we can essentially derive a new set of equations of motion, thought albeit non-linear ODEs, using Lagrangian Mechanics. My question is: how difficult is it usually to solve these non-linear ODEs? What are the usual numerical...

I curiously never had a problem solving Seperable Equations in the Seperable Equations chapter of the Boyce/Diprima book. I am the kind of person who likes to do things the long way, and encountered a problem solving for an Integrating Factor(Linear ODE, NOT EXACT) the long proofy way. I tend...

Homework Statement In control engineering, I want to have a mathematical model of a physical system as a set of input, output and state variables related by higher order differential equations. 2. Relevant concepts As we all know that, in control engineering, we can solve linear-system...

Homework Statement I'm given two equations for coordinates of a certain particle in the xy plane, \dot{x}+ωy=0 and \dot{y}-ωx=0. Then using the complex variable z=x+iy, find the differential for z, and solve it. Hence give x and y as functions of time. Homework Equations The...

Hello all, Next semester I will be taking a Network Analysis course in my EE degree. Moreover, we will be utilizing numerous mathematical concepts I have not yet seen. If anyone has (preferably free) access to any of the concepts to follow that they would be willing to share, I would be...