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

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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 ...
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35 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
16 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} + ...
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1answer
25 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
28 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
18 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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49 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
28 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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23 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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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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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
39 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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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 ...
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1answer
23 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
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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 ...
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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
25 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
29 views

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

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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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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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
24 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
28 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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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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2answers
51 views

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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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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Autonomous differential equation with periodic vector field

This is about introductory part in Chicone's text on Differential Equations: Suppose $F: \mathbb{R} \to \mathbb{R}$ is a smooth (continuously differentiable) positive function of period $p>0$. ...
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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 ...
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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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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
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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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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
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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 ...
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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 ...
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1answer
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What mathematics topics pertain more towards applied mathematics?

I'm entering my second year of undergrad (majoring in mathematics), and I've found that I am really bad at Linear Algebra, but very good at Calculus and Differential Equations. I'm hoping to venture ...
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Compute the operator norm of the linear transformation defined by the following matrix. [on hold]

Compute the operator norm of the linear transformation defined by the following matrix. \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}
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1answer
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Solution for ODE $\dot{x}=F(x)$ with $F:\mathbb{R}\rightarrow\mathbb{R}$ smooth, periodic & positive.

If $F:\mathbb{R}\rightarrow\mathbb{R}$ is smooth, periodic & positive and $x(t)$ is a solution for $\dot{x}=F(x)$ and $$T:=\int_{0}^{p}\frac{1}{F(y)}dy$$ then $x(t+T)-x(t)=p$ for all ...
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Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
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1answer
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How to prove that a differential equation has a solution

I want to prove that there exists $f : [0,1] \to [0,1]$ such that $$ w(y - f(y)) = \int_0^y g(x) dv(f(x)), $$ where $w : [0,1] \to [0,1]$ and $v : [0,1] \to [0,1]$ are continuous strictly ...
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4answers
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Solving this 1st Order PDE [on hold]

I am trying to solve the following PDE with an initial condition: $$u_x + u_y = x + y$$ with $$u(x, 0) = 0$$ I am not sure which method to use to solve this. Thanks
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A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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Showing a bound exists

I was able to derive the following differential equations I have to work with for a function $V$: $$ \begin{align*} dV(x_1,x_2,x_3,x_4) &= ...
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1answer
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Various forms of the Confluent Heun Equation

The Confluent Heun equation is expressed in various forms. It's non-symmetrical canonical form is: \begin{equation} ...
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1answer
47 views

Solution space of the differential equation $y'' + y =0$

To find the dimension of the solution space of the equation given $y'' + y = 0$ . Take $y=e^{mx}$Then we have to solve the equation $m^{2}+1=0$ for $m$ . Which gives $m=\pm i$. ...
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1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
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1answer
31 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
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1answer
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Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
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1answer
20 views

Where can I find the theorem that says an n order diffeq has n solutions?

I study engineering not mathematics and I feel my theoretical understanding of differential equations is SO wishy washy. I'm not interested in proving such a theorem at the moment, but I want to know ...
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36 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...