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

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Blow up solution of a Riccati's equation.

Consider the Cauchy problem $$ \left\{ \begin{array}{l} \dot x=x(t)^2+t\\ x(0)=0 \end{array} \right. $$ Show that its solution is not defined in $[0,3]$.
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1answer
26 views

Global existence of a dynamical problem.

Prove that all the solutions to the system $$ \begin{cases} \dot x= e^{-y^2}\sin(x^n+y^n), \\ \dot y= x^n\sin(x^n+y^n), \end{cases} $$ where $n$ is a fixed natural number, are defined on ...
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0answers
18 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
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1answer
34 views

Show that $f(x)$ satisfy the differential equation

Given a curve $C=\{(x,f(x)\in \mathbb{R}\times\mathbb{R}\mid x\in(r_1,r_2)\}$ with has the following property.(f(x) is $C^3$-function) At any point $(a,f(a))\in C$ if we change coordinate system by ...
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15 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
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1answer
44 views

solve this ordinary differential equation?

i have the differential equation $y'=\frac{y-x}{y-x+1}$, how i solve this? try: i tryed to substitute $u=y-x$, then $u=y-x\iff y=u+x\Rightarrow y'=u'+1$ then $y'=\frac{y-x}{y-x+1}$ become ...
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31 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
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1answer
24 views

Growth equations

Year 2003 there was approx. 10 % of a substance, the year 2013 the substance is 40 % One modell which can describe the substance speed of growth is that the substance, increases every moment, is ...
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26 views

Is ODE theory useful for developing numerical solvers for ODEs?

I will be doing research in developing numerical solvers for ODEs. I was wondering if knowledge of ODE theory will be useful and if so in what ways. I am asking because, I am inclined to take a ...
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Conditions for proper integrals with DSolve on Mathematica [migrated]

DSolve[{RCS'[s] == SNPH[s] - Sqrt[R2[s] - RCS[s]^2]/b,SNPH'[s] == RCS[s]/a^2, R2'[s] == 2 RCS[s] SNPH[s], RCS[0] == 0., SNPH[0] == snal, R2[0] == ri2}, {RCS, SNPH, R2}, s ]; Here {a,b,snal,r12} are ...
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25 views
+50

Calculating the constants in the general solution of second order homogeneous ODE reduced from Riccati equation

I am writing a code to simulate a kind of volumetric flow, and I have encountered the non-linear Riccati equation in its general form near the end of my calculations. I am having trouble finding the ...
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1answer
28 views

Optimization with a constraint given by a differential equation

I have the following differential equation $$\ddot\theta(t) = -k\sin{\omega t}\sin{\theta(t)} \quad \text{where} \quad \theta(0)=\theta_0, \dot\theta(0)=v_0$$ where $\omega$ is a known constant and ...
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11 views

Series solutions at regular singular points and Frobenius method using imaginary roots to indical equation

Consider the singular equation: $$ (1-t^2)y''-2ty'+12=0 $$ Find the regular singular points and solve the inical equation to obtain exponents of singularity $r_1$ and $r_2$. With $r_2\leq r_1$, look ...
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33 views

Boundary Layer, leading order, Pertubation Theory, Differential Equations

I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.: Find the leading order of the problem: $\varepsilon ...
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18 views

Which property can be used to derive a differential equation for a reparametrization

With $0\le t\le1$, two space curves given by: $$c_1(t)=(1,t,0)\quad\quad c_2(t)=(0,t,2t(1-t))$$ One of them, say $c_1$, must be reparametrized by $r(t)$ in order to minimize the area between the ...
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1answer
16 views

Express the differential equation that solves the below problem in polar form

Find the family of curves that intersect the lines $y=mx$ at a $45^{\circ}$angle I am able to solve this problem by expressing the differential equation in cartesian form $y' = ...
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1answer
30 views

solution of 3rd order non linear differential equation

I need help regarding solution of this equation which has been solved in a research paper but I cant figure it out. Please help $f^{\prime\prime\prime} + 3ff^{\prime\prime}- 3(f^{\prime})^2 - (Ha ...
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1answer
53 views

How can I solve this differential equation, what type is it?

How can I solve this differential equation, what type is it? $$(x^2+2x-2y)dx=dy$$ How can I find the integrating factor?
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32 views

Solutions to problems in the ODE book by Gerald Teschl

I am self learning ODE by the book: Ordinary Differential Equations and Dynamical Systems by Gerald Teschl. Anyone knows where I can solutions to the problems given in this book? Thank you.
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2answers
41 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
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1answer
8 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
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2answers
33 views

Is this PDE not solvable using characteristic lines?

My PDE book solved the following equation using the method of characteristic lines: $3u_x - 2u_y + u = x, u = u(x,y).$ I then encountered the following problem in the exercises: $u_x + u_y - u = 0$. ...
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28 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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1answer
46 views

$y'' + 4y = \sin^3(2x)$ Continuation of question

I have to find the solution for $y'' + 4y = \sin^3 (2x)$. We can use an identity to transform our equation to: $y'' + 4y = (3/4)\sin(2x) - (1/4) \sin(6x)$. Our guess for the particular solution ...
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0answers
16 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
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1answer
55 views

Is the following differential equation exact?

I have the following equation: $$ydx-(x+y^2)dy=0. $$ It seems to me the equation is not exact ($My=1, Nx=-1$), but then I don't know how to solve it. So is it exact or not? And if not, how can it be ...
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3answers
32 views

Techniques for solving coupled differential equations

I am trying to solve a system of coupled differential equations to plot streamlines using Matlab. The equations are these: $$\frac{\mathrm{d}x}{\mathrm{d}t} = -3x - 5y$$ ...
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2answers
33 views

Second order Non-exact ODE by integral factor

I'v solved the non exact ODE $$L[u] = xu'' + 2u' + xu = 0$$ by using integral factor sinx get $$u = \frac{Acosx + Bsinx}{x}$$ where A and B are constant if I want to solve $$xu'' + 2u' + xu = ...
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0answers
11 views

Existence for a differential inequality with mixed boundary conditions

For $\pi < \theta < 2\pi$, I have the following second-order differential inequality $$y''(s) (1 - \cos s - \tan \frac{\theta}{2} \sin s) + 2y'(s) (\sin s - \tan \frac{\theta}{2} \cos s) + y(s) ...
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2answers
117 views

Properties of solution of a differential equation

Consider the initial value problem $y'' = - y$ with $y(0) = 0$, $y'(0) = 1$, $0 \leq x \le \pi$. Without solving the differential equation, show that $y$ is symmetric about $m$, where $y'(m) = 0$.
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2answers
38 views

Is my solution of the following differential equation wrong?

I had to solve the following differential equation: $(x^2+y)\mathrm{d}x - x \mathrm{d}y=0$. The equation is not exact and so I solved it as a simple linear equation $$ ...
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1answer
90 views

Differential equation $x\dfrac{dy}{dx}-xy=y^2$

How do I rearrange this ODE so that I can use the basic ODE techniques to solve it i.e. separable, first order using integrating factor and/or Exact ODE. $x\dfrac{dy}{dx}-xy=y^2$
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3answers
42 views

Finding solutions of $y'''-4y''+5y'-2y=-x^2+5x+2$

Find all solutions of $$y'''-4y''+5y'-2y=-x^2+5x+2.$$ I know how to find the solutions of the corresponding homogenous differential equation $y'''-4y''+5y'-2y=0$. I've done that in the following way: ...
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2answers
23 views

The general solution for this differential equation?

Find the general solution of this differential equation: $$ \frac{dy}{dx} = \frac{3x^5 y^3}{4} $$ Here's what I've done so far: $ dy=\frac{3x^5 y^3 dx}{4} $ $ 4dy = 3x^5 y^3 dx $ $ \frac{4dy}{y^3} ...
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25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
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2answers
34 views

Undetermined Coefficients trouble

I need to solve $y'' + 4y = \sin^3(2x)$. However, I do not know what guess I would use for $g(x) = \sin^3(2x)$ since it is cubed. Usually, when $g(x) = b\sin(\beta x)$, I would give a guess for the ...
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1answer
52 views

Solve $x''-(\tan t)x'+2x=0$

Solve $x''-(\tan t)x'+2x=0$ knowing that $x=\sin t$ is one of the solutions I really have no clue how to do this one. The fact that one of the solutions is given makes it look a bit similiar to ...
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0answers
38 views

Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
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1answer
56 views

Solve $(x^2y^3+y)dx+(x^2y^3-x)dy=0$

$$(x^2y^3+y)dx+(x^2y^3-x)dy=0$$ So: $$\frac{\partial P}{\partial y}=3x^2y^2+1$$ $$\frac{\partial Q}{\partial x}=2xy^3-1$$ The question is how do I find the integrating factor when neither ...
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1answer
54 views

Showing $y_1$ or $y_2$ are not polynomials

proof that $y_1$ or $y_2$ are not a polynomial for any $n$ $$ y_1(x)=1-\frac{n(n+1)}{2!}x^2+\frac{(n-2)n(n+1)(n+3)}{4!}x^4-+\cdots$$ $$ ...
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1answer
36 views

First order ODE: $2tx'-x=\ln x$

$$2tx'-x=\ln x$$ First I differentiated both sides with respect to $t$ $$2x'+2tx''-\frac{x''}{x'}=x'$$ Then substituted $p=x'$ and changed it a bit $$p^2+2tpp'=p'$$ But I dont know what should I ...
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1answer
48 views

Deriving the equations of motion. Finding the critical points and determining their nature.

I have Duffing equation as $\ddot{x} - x + x^3 = 0$ Which I know describes the motion of a mechanical system in a twin well potential. I have let $y_1 = x, y_2 = \dot{x}$ and I want to derive the ...
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1answer
22 views

Last step in solving partial differential equation

I generally know how to solve these equations, but some of my last exercises has got me wondering, and I don't have the solution, just the answer. $$ y\frac{\partial f}{\partial x} - x\frac{\partial ...
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1answer
25 views

Green's function for periodic BVP

Problem. Determine the Green's function for the boundary-value problem \begin{align} y^{\prime\prime}(t)=&-f(t)\notag\\ y(-1)=&y(1)\notag\\ y^{\prime}(-1)=&y^{\prime}(1).\notag \end{align} ...
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1answer
45 views

Linearizing a nonlinear system of ODE about an equilibrium

Since the method below is probably correct, and correctness is potentially irrelevant to my ability to do what I want to learn. Assume below is correct. ...
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2answers
82 views

Solving a wave equation: $a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$

Find product solution $$a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ by the method of separation of variables So first off: $$u(x,t)=X(x)T(t)$$ ...
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1answer
39 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
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1answer
80 views

solution of $y' + y^2 = \varphi^2(x)$

I need to solve differential equation in the interval $[-\pi/2,\pi/2]$ \begin{eqnarray} y''(x) = y(x)\sin^2x \end{eqnarray} Trying $y(x) = \exp(\psi(x))$ yields, \begin{eqnarray} \zeta'(x) + ...
0
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1answer
34 views

Every solution of the system is attracted to the center manifold

I am trying to solve the following problem. Determine a center manifold for the rest point at the origin of the system \begin{align} \dot x &=-xy \\ \dot y&= -y+x^2-2y^2 \end{align} a) ...
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3answers
41 views

Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$

Let u(t) be a continuously differentiable function on [a,b] and the following inequality holds $\forall t \in [a,b]$ such that $u'(t) \leq f(t)u(t).$ Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$ I ...