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

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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
11 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 ...
2
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1answer
22 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 ...
3
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1answer
42 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 ...
3
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0answers
11 views

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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0answers
35 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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2answers
15 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 ...
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2answers
20 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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2answers
15 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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0answers
60 views
+100

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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2answers
105 views
+200

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, ...
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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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1answer
49 views

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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1answer
23 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 ...
2
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2answers
121 views

How to apply the Gronwall Lemma

Consider $x'=f(x)$ such that $(x_1,x_2)\mapsto(-x_1+2x_2,-2x_1-x_2)$. Show that for two solutions $x(t)$ and $y(t)$ of the above differential equation, we have: $$\lVert x(t)-y(t)\rVert \leq ...
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2answers
27 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 ...
2
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2answers
62 views

Differential equation : $y' = (x+1)/(xy+x)$

So, I have the following differential equation to solve : $$y' = \frac{x+1}{xy+x}$$ After several steps, I get here : $t^2 + 2t = 2x + 2ln(x) + c$ How do I isolate $t ?$ thank you! By the way, $ ...
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2answers
44 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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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
13 views

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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1answer
2k views

Numerically solving a system of nonlinear ODEs with boundary conditions

I have a system of 6 second-order nonlinear ODEs involving 5 different functions of a variable $t$. Every function has a boundary condition at $0$. I've never taken a differential equations class and ...
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0answers
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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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+150

Solving a Matrix DE involving the KL divergence

If we let $U_\mu$ be a vector field that associates a direction vector $U_\mu(\pi)$ with each $\pi \in $ unit simplex. Each such vector field is associated with a system of ODEs: $$ \pi'(u) = ...
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0answers
11 views

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
23 views

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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0answers
86 views

Blowing-up a singular point

I have this system of ODEs: $$x'=-y+ \mu x(x^2+y^2)$$ $$y'=x+ \mu y(x^2+y^2)$$ I already find that in $\mathbb{R}^2$ the only singular point is $(0,0)$. So I have to blow-up the singularity to find ...
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1answer
23 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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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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0answers
22 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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0answers
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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
293 views

Bernoulli Differential Equation of Second Order

How one can solve a Bernoulli differential equation of second order? i.e., solve the DE \begin{align} \frac{{d^2 y}}{{dx^2 }} + p\left( x \right)\frac{{dy}}{{dx}} + q \left( x \right)y = g\left( x ...
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1answer
24 views

Question Regarding a Second Order Ordinary Differential Equation

I was wondering if the solution to the following differential equation belongs to a class of special functions. If not, is it exactly solvable? \begin{equation} ...
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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 ...
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1answer
52 views

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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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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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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4answers
56 views

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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1answer
625 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
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1answer
22 views

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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1answer
69 views

Use of exclamation point

I'm quite puzzled by the use of an exclamation point in this paper. The authors introduce the following linear constraints to a quadratic program: $ \sum_k a^{(l)}_k b_j (\mathbf{x}_k) = r_j^{(l)} $ ...
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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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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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1answer
14 views

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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0answers
12 views

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
45 views

System of differential equation with variable coefficent

How to solve this system of differential equations $x'(t)=\frac{a+s}{(1-t)d}x(t)-\frac{b}{(1-t)d}y(t)$ and $y'(t)=\frac{a}{(1-t)d}x(t)-\frac{(s+b+(1-t)c)}{(1-t)d}y(t)$ where a,b,c,d and s are ...
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1answer
19 views

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
30 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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2answers
65 views

find general solution to the Differential equation

Find the general solution to the differential equation \begin{equation} \frac{dy}{dx}= 3x^2 y^2 - y^2 \end{equation} I get \begin{equation} y=6xy^2 + 6x^2 y\frac{dy}{dx} - 2y\frac{dy}{dx} ...
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2answers
141 views
+100

What is the value of $x$ such that $\frac{\text{d}^2y}{\text{d}x^2}=0$ where $\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$?

How can you find the values of $x$ such that $$\frac{\text{d}^2y(x)}{\text{d}x^2}=0$$ where $$\frac{\text{d}y}{\text{d}x}=-ae^{-bx}y-cy+d$$ with $$y(0)=y_0$$ and $$a,b,c,d>0$$ If it helps I can ...