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.
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...
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...
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...
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...