Diffeq Definition and 66 Threads

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)##?

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...

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...

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...

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...

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...

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=...

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...

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...

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...

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...

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...

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...

"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...

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.

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.

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...

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...

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...

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

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...

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...

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...

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...

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!

(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...

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...

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...

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...

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...

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...

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?

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?

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

\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...

[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...

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...

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...

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...