Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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Thickness of the Boundary Layer

Given an ODE $$\epsilon y''+2xy'=x \cos(x)$$ with boundary condition $y(\pm {\pi \over 2})=2$ Where is the boundary layer and what is the thickness of it?
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$(1-x^2) y^{\prime \prime}-2xy^{\prime}+12y=0$ [closed]

(1) Find all singular points and classify whether irregular or regular. (2) Find a power series solution about point $x=0$ (3) Write out first 4 terms of solution that have non-zero coefficients ...
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2answers
44 views

Integrate $ \int^{\pi}_{-\pi} (\pi^2-x^2)\sin nx \ dx$

Consider the function $f:(-\pi,\pi)\to\mathbb{R}$ be defined as $x \mapsto (\pi+x)(\pi-x)$ Compute the fourier series of $f$ So far, I've worked out $a_o$ by: \begin{equation} a_o = \frac{1}{\pi} ...
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2answers
22 views

Nonhomogeneous Euler-Cauchy equation $x^2 y''-2x y'+2y=4\log x$ [closed]

Determine solution of the associated homogeneous equation Use method of variation of parameters to find particular solution of non-homogeneous equation. State the general solution of the ...
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0answers
12 views

what is meant by “value of approximate numerical methods or method of digital computer” for finding solution of differential equation?

I was reading "motion against resistive forces" in Newtonian Mechanics: M.I.T. introductory physics series by A.P. French; here is the excerpt: [...] In general, the resistive force $\mathbf{R}$ ...
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1answer
43 views

Problems with a Simple Differential Equation

I am trying to solve the following: $y' = (y-5)(y+5)$ if $y(4) = 0$. So far, I have tried separating the variables and then use partial fractions and have followed these steps: (1) $A(y+5) + B (y-5) ...
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0answers
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Laplace transform: $y"+6y'+25y=100\sin(10000t)$ [closed]

Can't figure out how to do the following problem: Find $y(t)$ of $y"+6y'+25y=100\sin(10000t)$ $y(0)=5$, $y'(0)=10$
3
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1answer
77 views

Differential Equations in Milnor's Topology from the Differential Viewpoint

On page $23$ Milnor states: Let $\varphi$ : $\mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function which satisfies $$\begin{cases} \varphi(x) > 0, & {\rm for}\,\|x\| < 1 \\ ...
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4answers
74 views

What does d(something) mean?

In a book I am reading on differential equations, the author writes the following: $$e^{\int P(x) \mathrm{d}x}\mathrm{d}y+P(x)e^{\int P(x) \mathrm{d}x}y\mathrm{d}x=Q(x)e^{\int P(x) ...
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1answer
30 views

eigenfunction and eigenvalue

how can I find the eigenfunction for this BVP? $$y'' + (\lambda)\, y = 0, \,\, y(0)=0, y(2\pi)=0$$ with this case $\lambda >0$. I got the general solution: $$y(x)=c_1\cos(\sqrt{\lambda}x) + ...
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1answer
44 views

General solution of $(x^2-y^2)dx + (3xy)dy = 0$

Find the general solution to the homogeneous differential equation $$(x^2-y^2)dx + (3xy)dy = 0$$ The differential equation does not seem to be separable, and I'm having a tough time to put it in the ...
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1answer
38 views

Relatively simple system of nonlinear ODEs

There are a lot of questions like this on MSE as well as online resources on the subject, but a) the MSE questions are either unanswered or correspond to systems substantially different from this one, ...
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1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
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2answers
55 views

How to solve this 2nd order ODE

Consider $$\epsilon y''+yy'-y=0$$ with boundary conditions $y(0)=0$ and $y(1)=3$. I showed that the outer solution is $y_{in}(x)=x+2+O(\epsilon)$. Than for the inner solution, I wish to solve the ...
3
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1answer
75 views

Differential equation of a pendulum

Consider the nonlinear differential equation of the pendulum $$\frac{d^2\theta}{dt^2}+\sin \theta=0$$ with $\theta(0)=\frac{\pi}3$ and $\theta'(0)=0$. Using the series method, find the first four ...
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1answer
30 views

Find the solution to the initial value problem $\overline{x}' = A\overline{x}, x(0) = \begin{bmatrix}{2} \\ {28}\end{bmatrix} $

$A = \begin{bmatrix}{16/3} && {1/3} \\ {-64/3} && {32/3}\end{bmatrix} $ I got the general solution $\overline{x} = c_1 e^{8t} \begin{bmatrix}{1} \\ {8}\end{bmatrix} + c_2 ...
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1answer
17 views

Chebyshev differential equation solution

We have $\frac{d}{dx}((1-x^2)^\frac{1}{2}\frac{dy}{dx}) + n^2(1-x^2)^\frac{1}{2}y=0$ for integer $n\geq0$, and the question says to substitute $x=cosz$, which leads to $y''(z) + n^2y(z)=0$, and ...
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1answer
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ODE numerical solution [closed]

I tried it but ı cant create any code about this question.Does anybody help me.
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1answer
17 views

System of DEs with constant term

This is similar but not identical to standard examples in e.g. Paul's Notes, and while the math seems straightforward the results I get disagree with what I get from numerical simulation. Given a 2D ...
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2answers
20 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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4answers
45 views

$ y' = \frac{y}{\sqrt x } , y(0) = 2 $ Existence and Uniqueness?

I have the following first order ODE: \begin{equation*} y' = \frac{y}{\sqrt x },~y(0) = 2 . \end{equation*} Does there exist a solution? If yes, is it unique? How can I prove it by using existence ...
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0answers
23 views

Inital Value Problem from general solution

We have the following matrix: $\frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ And the inital condition: $\mathbf{Y_0} = (4,0)$ I have got the correct ...
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Second order pde problem.

A general second order partial differential equation of the form : $Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$ is solved by finding the characteristics equations by : $\frac{dy}{dx}$ = ...
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Boundary Value Problem for finding c1 and c2 [closed]

The general solution of x"+w(omega)^2x=0 is x(t)=c1 coswt + c2 sinwt satisfying: x(t0)=x0, x'(t0)=x1 x(t)=x0 cosw(t-t0) + x1/w sinw(t-t0)
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1answer
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Need help with this S-L problem

Consider the problem: $$\frac{\text{$\delta $u}}{\text{$\delta $t}}=\frac{\delta ^2u}{\text{$\delta $x}^2}\text{-u+x(1-x)}$$ The IC are given as: $$u(0,t)=1$$ $$u_x\text{(1,t)=0}$$ ...
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1answer
48 views

Tough Differential equation

Can anyone help me solve this question ? $$ \large{y^{\prime \prime} + y = \tan{t} + e^{3t} -1}$$ I have gotten to a part when I know $r = \pm 1$ and then plugging them into a simple differential ...
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0answers
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Uniqueness of Rectifying Coordinates: Question for Arnold's ODE Book

In section 7 of his book Ordinary Differential Equations, VI Arnold explains the `rectification theorem', that, given an ordinary differential equation $$\dot{\mathbb{x}} = \mathbb{v(x)}$$ where ...
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2answers
30 views

Differential equation

I know how to solve for particular solution when the right hand side is polynomials, but how do I these kind of differential equations when the right hand side isn't a polynomial ?
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1answer
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Interpreting d as an operator in differential calculus.

I am enjoying this mathematics book for the general public called Measurement by Paul Lockhart. For the most part, I am happy with his metaphors and intuitive explanations of the different concepts, ...
3
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1answer
30 views

Solve second-order linear ODE

everyone, I've a ODE to find a solution for it. The ODE is: $y''+2y'+10y=x^2e^{-x}cos(3x)$ I'm trying to solve it and find $y_h(x)=c_1e^{-x}sin(3x)+c_2e^{-x}cos(3x)+y_p$. The problem is to find ...
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1answer
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General Solution of ODE (complex eigenversion)

I am trying to figure out the general solution to the following matrix: $ \frac{d\mathbf{Y}}{dt} = \begin{pmatrix} -3 & -5 \\ 3 & 1 \end{pmatrix}\mathbf{Y}$ I got a solution, but it is so ...
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2answers
86 views

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$ [closed]

Simplify $(\cos x)^{2^{x^{\cos x}}}$ with respect to $x$ & $pi$... if $x > 0$ and $cos(x)$ $> 0$
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2answers
71 views

Differential equation with a function defined such that $f(x+1)=f(x)$

I just tried to do the following question, but I can't help but think I've done it completely wrong. The question states: The function $f$ satisfies $f(x+1)=f(x)$ and $f(x)>0$ for all $x$. ...
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0answers
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Find all Lie point symmetries generators for nonlinear equation [closed]

Find all Lie symmetries generators for nonlinear equation $$u_t=\frac {-1}{u_{xx}}$$
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Second Order Differential Equations… “Independent Variable Missing” Case - Justification

When trying to solve second order non-linear differential equations, "independent variable missing" case, we assume that we can express the derivative y' in terms of y... That is, the function ...
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1answer
15 views

Roots of Characteristic equation (ODE)

I have a question in regard to the following; considering the ODE $y^{(4)}+2y''+y=0$ we can factor and find that the roots are $$r_1=r_2=i$$ and $$r_3=r_4=-i$$ So, I thought that a solution will ...
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0answers
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How to solve this 2nd order, nonlinear ODE numerically?

How should I solve this second order, nonlinear ODE?: $$\left(\frac{f''(x)}{B}\right)^n=-(f(x)-a_0-a_1x-\cdots -a_mx^m)^n+A$$ Where $A,B>0$, $n$ is a large odd number and $m\approx 100$, and $f$ ...
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2answers
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Using substitution to make an equation into a separable differentiable equation

I have the question: By making the substitution $y = t^nz$ and making a cunning choice of n, show that the following equations can be reduced to separable equations and solve them. $$\dfrac{dy}{dt} = ...
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1answer
43 views

What is a differential equation?

Some definitions says a differential equation is a mathematical equation that relates a function with its derivatives Some say that it is just an equation involving derivatives of a ...
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2answers
30 views

exact equations: integrating factor

How do I go about finding the integrating factor for the equation: $$3x + \frac{6}{y} + (\frac{x^2}{y} + \frac{3y}{x})\frac{dy}{dx}=0$$ can I find it using the following method? $\frac{N_x - ...
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4answers
68 views

What is the result of $\frac{d\dot x}{dx}$?

I have a problem with a step in assignment. What is the result of $\frac{d\dot x}{dx}$ ? $x$ is the displacement, and $\dot x$ is the speed. I'm not sure if this equation itself is right. Thank you! ...
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2answers
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Help solving a first order non-linear differential equation derived from the navier-stokes equation

I am an engineer studying an unsteady-state flow through a pipe. The transient Bernoulli equation of this system, which I picked up from here ...
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1answer
20 views

Euler's explicit differential equation solving method demonstration

Could someone help me to demonstrate the Euler's method on its explicit form ? Im having a few problems, because I need to start directly with a second order equation : $y" (x)=f(t,y(t))$ thanks :) ...
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0answers
42 views

Differential equations, Dynamical Systems, and an introduction to chaos [closed]

Maybe this question doesn't belong here but i'm starting on a new book and it would really help if i had a solution manual of it. The book is 'Differential Equations, Dynamical Systems, and an ...
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0answers
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Qualitative theory of ODE & Difference equation [closed]

Derive the compound interest formula by using difference equation and show that for compounding the formula reads as and the difference equation converted to differential equation
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1answer
33 views

Setting up and solving a system of differential equations for account balance [closed]

Liam opens a bank account with an initial balance of $1000$ dollars. Let $b(t)$ be the balance in the account at time $t$. Thus $b(0)=1000$. The bank is paying interest at a continuous rate of $5$% ...
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2answers
37 views

Change of variables in multivariable differential equations

This is a very easy question about how to justify the change of variables. Let $f$ be a $C^1$ function of two variables $x,y$. Introduce the variables $s,t$ as: $$\begin{cases} s=x+y \\ t=x-y ...
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3answers
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Gaining some insight about Picard–Lindelöf theorem.

In class I have been introduced to the Picard–Lindelöf theorem. It was written down in all its technical glory.Now: the important things to remember from it were that if a function is continuous; and, ...
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0answers
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is 1/x a lipschitz continuous function? [closed]

Actually, I would like to know if the dynamics associated, with initial condition $x(0)=1$, has any solution in $\mathbb{R}^2$. I tried to find the solution and I got that $x=\pm\sqrt{2t+1}$, but then ...
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0answers
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Differential Partial equation with fourier transform [closed]

how to apply the Fourier transform to the following problem $P:y''+4y=0$ with boundary conditions $y(0)=0=y(\pi)$