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

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monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
2
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1answer
25 views

the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$

Find all possible $(\lambda,u)$ where $\lambda \in \mathbb R$ and $u\ne0$, to the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$ My Effort: for ...
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1answer
29 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
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1answer
28 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...
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1answer
68 views

Solving a Linear ODE

Solve the following linear ODE $$3t^{2}y'+t^{3}y=cos(t)$$ What i tried Since this is a linear equation, i used the integrating factor method. First i didide both the LHS and RHS of the equation by ...
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27 views

Example: Solve a Second Order Nonhomogeneous ODE with Constant Coefficients by Variation of Parameters (2R-17)

Problem to solve: $$(D^2-2D+1)y=\frac{e^x}{x^3}$$ Answer in text: $$y=(c_1+c_2x)e^x+\frac12\frac{e^x}{x}$$ Our solution begins by rewriting the ODE in a more familiar form: ...
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1answer
38 views

Study of systems of Linear Differential Equations?

Is there any area of mathematics that deals with and formalizes systems of Linear DEs, akin to how Linear Algebra deals with systems of linear equations? Does it provide any insightful results?
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1answer
31 views

Linearity of a differential equation

The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one: $a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + ...
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1answer
3k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
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35 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
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36 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
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1answer
42 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
2
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1answer
39 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
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1answer
42 views

Why do we need sturm liouville form to solve ODE?

What is the reason that we have to recast a 2nd order ODE into SL form to find its eigenfunctions? for example, let $Ly=y''+y'+\frac{y}{4}=-ky$, boundary conditions $y=0$ at $x=0$ and $y-2y'=0$ at ...
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0answers
35 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
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1answer
33 views

Pass the lower limit to $-\infty$ for an integral of positive function

Hello I have an very elementary calculus problem. Let $\phi(\eta)$ be a real value function satisfying \begin{equation} \phi(-\infty)=1,\quad \phi(+\infty)=0, \end{equation} Let $g$ be a positive ...
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1answer
49 views

Solving a Variable Separable Differential Equation

The equation is $$y'=\frac{1}{18}x(81-y^2)$$ with $y(0)=81$, and I have to solve for an equation of the form $y(x)$ So I do $$\frac{dy}{(81-y^2)}=\frac{1}{18}x \ dx$$ I integrate both sides, and ...
2
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1answer
41 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
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0answers
17 views

Modulus of Green's function

Consider the nonlinear differential equation $$y'' = f(x,y,y')$$ together with the boundary conditions: $y(\alpha) = A$ and $y(\beta) = B$. Now $y(x)$ is a solution of this problem if and only if ...
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1answer
63 views

Find roots of $ω^x+(ω^x)^2+1=x$ [closed]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
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1answer
43 views

Solving an equation by Laplace transform

Consider the following equation: $$ y^{\prime\prime}(x) +x = \int _0 ^x (x-u)y(u)du \qquad y(0)=0 \quad y^{\prime}(0)=1$$ I solved it by Laplace transform and got $-\sinh x$ as a solution. It is ...
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1answer
22 views

Extracting differential equations [duplicate]

$$\frac{dx}{dy} = \frac{x(\alpha - \beta y)}{y(\delta x - \gamma)}$$ How do I extract two differential equations (y as a function of x and x as a function of y) from the equation above? I could ...
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2answers
52 views

Solving a first order ODE

Consider the initial value problem $$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$ (a)Determine how the solution behaves as $t$ tends to infinity. (b)If $y_{0}=2$,find the time $T$ at which the solution ...
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0answers
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Particular Solution of ODE 2?

today I found an interesting example considering viscoelasticity. While I was solving the given ODE I wondered, how the authors came to the solution they gave ... (the thing I am talking about is ...
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1answer
25 views

Second order differential equation, physics.

I need your input on this exercise I'm doing: "A 2-kg mass is suspended from a string. The displacement of the spring-mass equilibrium from the spring equilibrium is measured to be 50 cm. If the mass ...
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0answers
17 views

Find a Lyapunov function

How can I find a function of Lyapunov ? is there specific methods ? For exemple, how can I find the Lyapunov function of ...
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2answers
38 views

Where is f and gnot differentiable? [closed]

is my answer correct ???? I try to solve it but not sure if correct or not please help
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0answers
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How can I solve this differential equation with fourier series?

Find a formal solution $u(x; y)$ by using Fourier series. (Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with ...
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0answers
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Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
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1answer
36 views

Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ [closed]

Find the Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ satisfying the condition $y(0)=0, y(4)=3.$
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1answer
53 views

How to solve $y'''-y=x+1$?

Solve ODE: $$y'''-y=x+1$$ To find the particular solution, I thought to impose $$y_p(x)=ax^3+bx^2+cx+d$$ Fair Enough? Or should I consider other?
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1answer
18 views

Solving first order discrete differential equation

I have a question about solving a first order discrete differential equation. The equation is $x' = Ax$ with $x_{0} = x0$ I found Runge Kutta could solve the differential equation, but required ...
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0answers
28 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
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1answer
48 views

Prove that there is at most one function that verifies

let $f$ be a function defined on $\mathbb R$ of class $C^2$ and $g$ is a function of class $C^2$ Prove that there is at most one function that verifies $$g(x)=f(x)+\int_{0}^{x} (x-t)g(t) \, ...
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1answer
14 views

Can a first-order autonomous differential equation have a single steady state at x=0 that is not approached exponentially fast?

I know that x' = kx is exponential growth. I've tried to come up with some solutions. My first solution is x' = -x^3, which has an asymptotically stable point at x=0. It is approached more slowly, ...
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4answers
69 views

Solving a homogeneous system of three ODEs with variable coefficients.

I am interested in solving the following system of ODEs: $$ \begin{pmatrix} x'(t) \\ y'(t) \\z'(t) \end{pmatrix} = a \begin{pmatrix} 0 & -B_2 & B_1 \sin \omega t \\ B_2 & 0& -B_1 ...
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1answer
31 views

How to find general solution of PDE

How to find general solution of equation $$ U_{xy}- \frac{U_x}{y} =0 ? $$ My approach: $$ U_{xy} = \frac{U_x}{y}. $$ Integrate w.r.t $x$ $$ y \ U_y = U + c $$ integrate w.r.t y I don't know how to ...
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1answer
32 views

Heat Equation, possible solutions

NOTE: This is a homework problem. Please do not solve. I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the ...
2
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1answer
38 views

Intuition to solving partial differential equations

I do not understand how to solve PDEs using the geometric method. I just do not understand the logic behind the solution. For example, the constant coefficient equation $$au_x + bu_y = 0,$$ where a ...
2
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1answer
49 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
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1answer
50 views

What Did I Do Wrong When Solving For This 2nd Order Differential Equation? (answered myself)

$$ \frac{y''}{y'}+y' = f(x) $$ I set the following to be true: $$ y = \sum_{n=0}^{\infty} a_n x^n $$ $$ f(x) = \sum_{n=0}^{\infty} b_nx^n $$ Therefore: $$ y'' = y'(f(x)-y') $$ $$ ...
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1answer
29 views

Show differential equation $y(y^2+2x)+2x(y^2+x)\frac{dy}{dx}=0$ has integrating factor of form $\mu=x^2y^k$ Find the general solution to this equation

I really dont know where to start this one. Ive been going round in circles just trying to simplify the equation... Any tips on how to start this? Thanks!
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1answer
15 views

Find the general solution of the following first order differential equation $\frac{dy}{dx}=\frac{8x^3+3x^4}{y^4}$

I have multiplied both sides by $y^4$ which gives me $\frac{dy}{dx}y^4=x^3(3x+8)$ Then do I integrate both sides with respect to x? $\int\frac{dy}{dx}yx^4dx=\int{x^3(3x+8)dx}$ Am I still on the ...
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0answers
26 views

Lipschitz Constant (Burden and Faires Exercise)

There's an exercise in Burden & Faires Numerical Analysis book, Section 5.1 #2a, where they appear to want the reader to verify that a Lipschitz constant exists for the following ODE: ...
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1answer
44 views

Finding a strict Liapunov finction

I need to find a strict Liapunov function for this system at the equilibrium point $(0,0)$ $$x'= -2x-y^{2}$$ $$y'=-y-x^{2}$$ Also need to determine $\delta > 0$ as large as possible so that the ...
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2answers
37 views

How to solve this special second-order non-linear ODE?

I'm in trouble with solving the following differential equation: $0=y\cdot y''-y^2-2(y')^2$ I would be thankful for any help!
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Importance of Initial Guess in the numerical solution to the following fluid flow problem

Greetings Stackexchange community. Forgive me if the question is repetitive and/or answered before. I am currently working on a simple fluid flow problem, 'Heated laminar vertical Jet'/Brand and Lahey ...
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19 views

How to solve an ODE with Dirac delta multiplicative and additive terms?

I'm trying to solve the following ODE: $\frac{dx}{dt}=\beta x^2+(\beta+\delta (t-a))x+\delta(t)$ where $\delta(t-a)$ is a Dirac delta function. $x(0)=x_0$ I've tried Laplace transform, the Lagrange ...
0
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1answer
28 views

Division of differential equations

$$\frac{dx(t)}{dy(t)}=\frac{\alpha x(t) - \beta x(t) y(t)}{-\gamma y(t) + \delta x(t)y(t)}$$ How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.
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2answers
57 views

A boundary value problem for the ODE $y''+y'/y=-1$

Solve the Ode: $$ \begin{cases} \frac{d^2y}{dx^2}+(\frac{1}{y})\frac{dy}{dx}=-1 \quad \hbox{for $0<y<1$} \\ y(1)=y'(0)=0. \end{cases} $$ Putting $\frac{dy}{dx}=p$ then ...