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

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Find $\varphi_{1}$ from $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$

I have $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$ where $q_{1}=0$ and $\dot q_{1}=v_{0}$ and I must find $\varphi_{1}$. I know that $\varphi_{1}$ must be zero but I must demonstrate it first. ...
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1answer
30 views

How to solve differential equation $3p^2e^y-px+1=0$ ,$p =\frac{dy}{dx}$

How to solve differential equation $$3p^2e^y-px+1=0$$ where $$p =\frac{dy}{dx}$$ I have tried to solve for p and for x, but i am not getting anywhere. Can someone help me with this Thanks
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1answer
45 views

Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem $y'=y^2$, $y(0) = 1$, exist for all real $x$. (b) Find a solution of the initial value problem in (a). On what interval does it ...
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1answer
76 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
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22 views

Determine for what values of m the function is a solution

I was working through some differential equations and came across this problem. Determine for which values of $m$ the function $\phi(x)=e^{mx}$ is a solution to the given equation. A) ...
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11 views

Solve the following differential equation subject to the specified boundaries:

Solve the following differential equation subject to the specified boundaries: my answer please review my answer and correct it if it is wrong thanks
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1answer
22 views

Trigonometric Differential Equation 3

$(x\cos y-y\sin y)dy+(x\sin y+y\cos y)dx=0$ ATTEMPT: Rearranging the terms: $(x\cos ydy+y\cos ydx) -y\sin ydy+x\sin ydx=0$ Dividing by $\cos x$ we get: $(xdy+ydx)-y\tan ydy+x\tan ydx=0$ $ ...
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1answer
21 views

Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
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1answer
32 views

Find the position function from the piecewise-defined velocity function

I am getting stuck on a position function problem in my Diff Eq class. Problem 22 is shown on the right in the picture below. On the left is the answer. My work below shows that I get stuck ...
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16 views

Study materials for Differential equations and Fourier analysis

In two days, on Monday, a new course called Introduction to differential equations starts, and when that ends in one month another called Fourier analysis and its application starts (Both are actually ...
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5answers
93 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
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2answers
124 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
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1answer
34 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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5answers
97 views

Solve $y''=y^2$

Are there any 'basic' solutions to this differential equation (ie using polynomials, exponetials, trigonometric functions and logarithms)? I cannot figure it out at all using the techniques I know for ...
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1answer
28 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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1answer
43 views

Number of solutions of the differential equation ${dy}\over {dx}$=$y^{1/3}$ $y(0)=0$

The given differential equation is ${dy}\over {dx}$=$y^{1/3}$, $y(0)=0$ I got the solution $$y^{2/3}={{2}\over {3}}x$$ $$i.e. y^{2}={{8}\over {27}} x^{3}$$ $$i.e. y= \pm \sqrt{{{8}\over ...
2
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1answer
46 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
2
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1answer
83 views

Exact Differential Equations

$M(x,y)dx + N(x,y)dy=0$ is said to be a perfect differential when $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$. Let $M_y=\frac{\partial (M(x,y))}{\partial y}$ and ...
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1answer
36 views

Sketching phase portrait of an ellipse

I have a system of linear ODE's as follows: $$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$ which has solution $$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + ...
2
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1answer
37 views

Taylor Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
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2answers
44 views

Slope field of $y'=x^2 - y^2$

I don't know how I am supposed to go about creating a table with slope values for the graph so that I can sketch them. I knew how to do it when $y'$ equations had $y$ only or $x$ only, but not when ...
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1answer
19 views

Definition of `equivalent systems of linear differential equations'

I'm reading F.Beukers' `Notes on differential equations and hypergeometric functions', and I can't work out the details of something that seems obviously true. We have a field $K$ endowed with a ...
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1answer
40 views

Why is t used instead of delta t?

Consider a tank that holds $V$ liters of water. Let $x_0$ kg of salt be dissolved in the water at time $t_0$. Suppose that $V_o$ amount of the mixture is leaving the tank in every time interval, ...
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0answers
52 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
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1answer
31 views

Topics to master (be literate at) before differential equations?

Good evening, I'm really enthusiastic about learning differential equations because it was said that D.E. is the most important tool of mathematics "can be used for modelling real-world physical ...
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1answer
41 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
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0answers
31 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
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0answers
19 views

Finite difference for variable coefficient with Neumann Boundary

The equations is the same as this post, but with respect to the Neumann boundary. The physically correct boundary conditions for this equation are \begin{equation} A(x)\frac{\partial u(x)}{\partial ...
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2answers
41 views

Homogeneous 1st order ODE

This question comes from Schaums Calculus, CH59 Q18 which has had me confused for a couple of days now. Solve: $$ {dy \over dx} + y = xy^2 $$ I understand that this is a non-linear first order ode, ...
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0answers
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Green's Functions: Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
2
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1answer
25 views

Finding an equilibrium solution to a first order system of equations.

Given a model: $ y''+\alpha y'+\beta y + \gamma y = -g $ I can see that it can be converted to a system of first order equations as follows: $y_{1}=y$, $y_{2}=y'$ and as such $y_{1}'=y'$ and ...
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2answers
35 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
2
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2answers
19 views

Help interpreting the solution for a differential equation

The differential equation is $\frac{dx}{dt} = x + x^2$ Solving for $x$, I got $x = (ce^t)/(1- ce^t)$ where, $c = x_0/(1+x_0)$ and $x_0$ is the initial value of $x$ at $t=0$ Now, the value of ...
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2answers
32 views

Differential Equations Pressure and Density derivation

The pressure $p$, and the density, $\rho$, of the atmosphere at a height $y$ above the earth's surface are related by $dp = -g \rho\; dy$. Assuming that $p$ and $\rho$ satify the adiabatic equation of ...
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1answer
35 views

Show that $m_1=\frac{m_2-\tan{(a)}}{1+m_2\tan{(a)}}$ [closed]

Given: $m_1=\tan{(a_1)}$ and denotes the slope of the required family at some $(x,y)$ $m_2=\tan{(a_2)}$ and denotes the slope of the given family at the same $(x,y)$ it also gives the hint that ...
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1answer
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Solving Differential Equation $\frac{dy}{dx} = 1 -\sin(x+y)/(\sin y \cos x)$ by separating variables

Initial value is $y(\frac{\pi}{4})$. I got to $\frac{\mathrm{d}y}{\mathrm{d}x} = 1 - \frac{\sin(x) \cos(y) + \sin(y) \cos(x)}{\sin(y)\cos(x)}$ by using the $\sin(x+y) = \sin(x) \cos(y) + \sin(y) ...
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0answers
37 views

Diagonalization: Differential Equations

The booking being used for this course is Differential Equations and Dynamical Systems by Lawrence Perko. The problem is as follows: Let the $n\times n$ matrix $A$ have real, distinct ...
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1answer
26 views

Uncoupled Linear System: Differential Equations

I'm trying to make sense of a problem I was given in class and I want to know if I am on the right track. The question is as follows: If $\vec{u}(t)$ and $\vec{v}(t)$ are solutions of the linear ...
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2answers
35 views

How can I find the differential equation for a (R+L)||C circuit?

I have a question about a parallel series RLC circuit; the capacitor is parallel to the {inductor + resistor}. The capacitor is charged at an initial voltage $U_{C,0}$ and the inductor has initially ...
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22 views

how can I prove that a derivative of an implicit function is bounded?

I have the following implicit function $V(\tau,\mu)$. The function is bounded and continuous and differentiable on $\mathbb{R}$. What other properties or assumptions should I make or what conditions ...
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1answer
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fluid dynamics in polar coordinates

On page 12 of Malham's fluid dynamics notes the following flow field is considered: $\boldsymbol u= (u,v) = (kx, -ky)$. It's easy to see in these Cartesian coordinates that this is solenoidal: ...
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How to prove that $J_\frac{-5}{2}(x)= \frac{\sqrt2}{\sqrt{x\pi}}[\frac{3}{x}\sin x+\frac{3-x^2}{x^2}\cos x]$

How to prove that $$J_\frac{-5}{2}(x)= \sqrt{\frac{2}{\pi x}}\left(\frac{3}{x}\sin x+\frac{3-x^2}{x^2}\cos x\right)$$ I want to do this by using the definition of $J_{-n}(x)$ then putting value of ...
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0answers
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If the solution of the following ODE unique with given initial value?

I am considering the following ODE: $$t\frac{d}{dt}f(t)=F(f,g)$$$$t\frac{d}{dt}g(t)=G(f,g)$$. F,G are polynomials. For given an initial value $f(0)=f_*,g(0)=g_*$ satisfying ...
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1answer
47 views

Why is the solution of an ordinary differential equation required to be defined on an interval?

I am reading A First Course in Differential Equations with Modeling Applications (10th Edition) and here is a definition: Any function $\phi$, defined on an interval $I$ and possessing at least ...
2
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0answers
38 views

Stability of an equilibrium

From a Center-Manifold reduction I get the following system: $$ \begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}-y(2x^2-2xy+y^2)\\x\end{pmatrix} $$ The aim is to analyze the stability of ...
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1answer
26 views

Finding the Equations of Motion for the Leapfrog Integrator

I understand that the Leapfrog Integrator is used to find an integral for Newton's Laws of Motion and that the Equation of Motion are given by: $$\frac{dx}{dt} = v$$ and $$\frac{dv}{dt} = F(x) = ...
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1answer
30 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
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0answers
27 views

How to solve diff. eq. involving function taking expression including itself as variable

$$ f(x)=f \left( x \pm \frac{l}{\sqrt{1+\dot{f}^2}} \right) \mp \frac{l}{\sqrt{1+\dot{f}^2}}\dot{f}^2, $$ where $l$ is a constant. How is such a beast even approached? If anyone got intuition for ...
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1answer
25 views

Linear Systems: Differential Equations

The book being used for the course is Differential Equations and Dynamical Systems by Lawrence Perko. The question is as follows: Find the general solution of the linear system ...
0
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0answers
37 views

Where can I find this definition of “expected value”?

I need bibliography or some text about this definition: "Define the expected value of a function by: $E_{t}(x(t))=(\frac{1}{t})\int_{0}^{t} x(s)ds$. " I think that it's statistics or functional ...