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

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Show that the function is positive

Let $f:\mathbb R\to\mathbb R$ be a Lipschitz continuous, monotone increasing function, with $f(0)=0$, if a function $\phi$ satisfies; $\displaystyle\cases{ \phi'(t)=f(-\phi(t))-f(\phi(t)) ...
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Get a special form of an linear System of ODE (using polar form)

In this post Converting an ODE in polar form it is shown that a linear system of ODE $$ x'=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ can be written in polar coordinates ...
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1answer
29 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
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1answer
18 views

Converting an ODE in polar form

Convert the ODE system $$ \dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x $$ into polar form. You should get two equations $$ \frac{d}{dt}\Phi(t)=...\\ ...
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7 views

What are some planes (spaces) akin to the trace-determinant plane in other disciplines?

When studying basic differential equations, I found the trace-determinant plane incredibly illuminating. Similarly, I find it very helpful to see different kinds of conics as slices of a cone. What ...
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1answer
13 views

System of Linear differential equations with variable coefficients

Could someone please suggest a technique for solving the following linear system of ODEs: $$ \begin{array}{l} i\alpha \frac{{dx(q)}}{{dq}} = \left( {\beta - 2c\cos (q)} \right)x(q) - ig\,y(q)\\ ...
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Prove $x\to 0$ as $t\to \infty$ if we consider the system of equations $x'=(A+B(t))x$ where $B(t)\to 0$ and $A$ has negative eigenvalues.

Consider a matrix $A$ such that all of its eigenvalues are negative. Consider $B(t)$ where $B(t)\to 0$ as $t\to\infty$. Then consider the system of equations $$ x'=(A+B(t))x$$ Prove that any ...
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1answer
13 views

Solution of a second order ODE

I want to solve the following ODE \begin{align} f^{''}(t) + \frac{1}{t} f^{'}(t) &= 0 \\ f(1)&=0 \end{align} Is this an Euler-type ode? In oder to find the solution, i have to rearrange the ...
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0answers
20 views

Solve the following ODE

Solve the following ODE $$(y-x)\left(1+x^2 \right)^{\frac{1}{2}}\dfrac{\mathrm{d}y}{\mathrm{d}x}=n\left(1+y^2 \right)^{\frac{3}{2}}$$ I have tried substituting $y=\tan \theta$ and $x=\tan \phi$ ...
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Nonhomogeneous DE with constant coefficients , reduction of order [on hold]

Please solve this equation using the Reduction of order method .. $$ y''-4y' + 3y = x $$
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If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
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2answers
35 views

A strange 3rd order ODE

This is the original ODE: $ y^{1/2}y'''+e^{-x}(y'')^{2+c}-(\frac{xy}{x+1})y'=x $ with c is a positive number. $y(0)=1,y'(0)=0,y''(0)=1$ $1st$ question: If x is large, then $ y^{1/2}y'''$ and ...
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35 views

Prove that solutions to linear system form a vector space of dimension $\geq 2$

I accept & appreciate any form of help with the following problem: $B_{nxn}$ "periodic matrix" with period $T$ such that $B(t+T) = B(t)$ for all $t\in \mathbb{R}$. Assume that the system $x' = ...
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24 views

Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
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3answers
79 views

If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...
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22 views

Changing form of differential equation

How would you change the differential equation: $3xy^{''}+y^{'}+12y=0$ into a form where the coefficient of the first term is 1? leaving just $y^{''}$ I should probably know how to do this, but I'm ...
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1answer
25 views

Limit of a continuously differentiable function that statisfies

Let $x(t)$ be a continuously differentiable for all $t>0$, and such that: $$\lim_{t\to \infty}[x'(t)+x(t)]=\alpha$$ I need to show that $\lim_{t\to \infty}x(t)=\alpha$ My goal is to show that ...
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1answer
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Proving that the solution to the IVP exists given a condition on the right hand side

Consider the following IVP: $x'=f(t,x)$ $\ $and $\ $ $x(0)=x_0$ where $x\in \mathbb{R^n}$ and $t\in \mathbb{R}$. Suppose that for all $(t,x)\in\mathbb{R^{n+1}}$: $|f(t,x)|\leq b(t) |x|^2$. In order ...
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Interval of solutions of this differential equation

For the following differential equation initial value problem, $y' = \frac {-t + (t^2 + 4y)^{1/2}} {2}$, $y(2) = -1$ the interval of t for which the solution $y_1 = 1 - t$ is valid is $t \ge 2$ and ...
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21 views

Limit of Solution of an ODE

Consider the following differential equation: $y^{'}(t)=g(y)$ where $g$ is a continuous function from $\mathbb{R^n}$ to $\mathbb{R^n}$. Assume that $y(t)$ is a solution to the previous ODE. Suppose ...
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1answer
19 views

Projectile Motion using cos and sin theta???

Golfball is struck to clear a tree 20m away and 6m high at an angle of elevation of 40degrees. Find the speed of the ball when it leaves the ground. I've created my displacement equation with i and j ...
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3answers
32 views

How to find inverse laplace transform

$$ F(s) = \dfrac{6s+9}{s^2-10s+29} $$ How do you solve the inverse Laplace transform of this above equation?
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1answer
19 views

Why existence of Lyapunov function implies Lyapunov stability at the equilibrium point

Why existence of Lyapunov function (locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite) implies Lyapunov stability (i.e for any ...
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1answer
27 views

How to solve Laplace initial value problem

$$ y''+36y = f(t) $$ $$ f(t) = \begin{cases} 1, & \text{0 ≤ t < 8} \\ 0, & \text{8 ≤ t < ∞} \end{cases} $$ $$ y(0) = 0 $$ $$ y'(0) = 1 $$
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19 views

On a cauchy problem (differential equation system) .

I am approaching the theory of these kind of problems but I am missing an example. I am tasked to solve: $$X'= \left( \begin{array}{ccc} 1 & 4 \\ 1 & 1 \\ \end{array} \right)X + \left( ...
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1answer
25 views

Solution to Differential Equation $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$

I'm trying to solve the following Differential Equation: $\left( 1-2\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)+2 \lambda h(x,y,z)=0$ The unknown function is $w(x,y,z)$. The ...
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the solution to $1$st order linear nonhomogeneous equation (leading coefficient may vanish)

I'm working on the $1$st order linear nonhomogeneous equation $c_1(x)y'+c_2(x)y=f(x)$. I know the homogeneous solution is $h(x)=e^{-\int_ a^x c_2(t)/c_1(t) dt}$ and then $y(x)= e^{-\int_ a^x ...
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1answer
42 views

Specific form of differential equation

Suppose the function $$ f(x)=p(x)\,\mathrm{e}^{q(x)}, $$ is the solution to a differential equation. From which family of differential equations would $ f $ arise?
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Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
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14 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
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25 views

Convergence to a fixed point

When the following system is given: $x(k+1)=r-rx(k)$ where $r>=0 $ is a parameter Can someone explain why the fixed points are given by: $x(k+1)=x(k)=x^*$, so $x^*=\frac{r}{1+r}$? and how to ...
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2answers
24 views

Second Order Linear Differential Equations

The question is that find the general solution of differential equation y"-2y'+y=e^x i know that y(c)=Axe^x + Be^x then let the f(x)=e^x , so y=pxe^x as f(x) is in the complementary function. so ...
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2answers
33 views

$\lim\limits_{t\to\infty}t-x(t)=0\ ?$

Let $\displaystyle\cases{ x'=\frac{t-x}{1+t^2+x^2} & \cr x(1)=1 }$ be the Initial value problem, prove or disprove $\lim\limits_{t\to\infty}t-x(t)=0$ We've already proved that: for $t>1, ...
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1answer
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Help with Implicit Differentiation: Finding an equation for a tangent to a given point on a curve

When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book. The problem is ...
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2answers
26 views

Find the limit and differential equation

We have the following equality: $$ f(x + \Delta x) = f(x) + a \Delta x \, f(x) - 10 \, b \Delta x $$ with a & b constants. If we take $\lim_{\Delta t \to 0}$ , we get a differential equation. My ...
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1answer
10 views

Second-order Van der Pol to First Order System

I'm trying to figure out how my professor arrived at the following first order system for the Van der Pol equation $x''+ c(x^2-1)*x' + x = 0$. It's supposed to be equivalent to the first-order system ...
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How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
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1answer
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Solution to differential equation $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$

I'm trying to solve the following differential equation: $\left( 1-\lambda\frac{\partial}{\partial z}\right)w(x,y,z)-g(x,y,z+h)=0$ here $g(x,y,z+h)$ is a known function that however i will leave ...
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1answer
15 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
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A basic doubt on Lyapunov function

Let $V$ be a Lyapunov function. Let $\epsilon > 0$ be any number such that the set $|y| < \epsilon$ is in the open set on which $V$ is defined. Then Hartman claims the following : For any ...
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1answer
27 views

Solving the ODE $(x^2 - 1) y''- 2xy' + 2y = (x^2 - 1)^2$

I want to solve this ODE: $$(x^2 - 1)y'' - 2xy' + 2y = (x^2 - 1)^2.$$ I found out that $y_1 = x$ and $y_2 = x^2+1$ are solutions of the associated homogeneous equation, $x$ by inspecting, and ...
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17 views

Periodic solutions and critical points

I was going through a lecture, and for an ODE: $x' = x(5-x-2y), y'=y(-6x+x+3y)$ Which has critical points at : $(0,0) (0,2) (3,1) (5,0)$ My professor posed the question as to why the periodic ...
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1answer
30 views

Chain rule and x =cos(θ) substitution on the Legendre ODE

I am having difficulties with a question where I am required to use the chain rule, and then use the substitution x = cos(θ), on the Legendre differential equation, which is $$(1-x^2)y'' - 2xy' + ...
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3answers
50 views

Solve differential equation, $x'=x^2-2t^{-2}$

Solve differential equation: $x'=x^2- \frac{2}{t^2}$ Maybe is it sth connected with homogeneous equation? I have no idea how to solve it.
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3answers
37 views

Solve $2xdy+(x^2y^4+1)y dx=0$ [on hold]

Solve differential equation: $2xdy+(x^2y^4+1)y dx=0$ Hint: use: $y=zg(x)$ Any ideas?
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3answers
72 views

Solve y''' = y with given conditions?

I'm given the differential equation: $$y''' = y$$ which solves to: $$y(x) = c_1e^x + e^{-x/2}\left(c_2\cos\left(\frac{\sqrt{3}x}{2}\right) + c_3\sin\left(\frac{\sqrt{3}x}{2}\right)\right)$$ But I'm ...
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15 views

How to find the solution of second order equation

I have a question that need your help. Given a $x(t)$ is a second-order Markov process: $$\frac {dx}{dt}=x_2+\omega$$ $$\frac {dx_2}{dt}=-2x-2x_2$$ with white noise $\omega$ is $(0,2)$. How to find ...
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1answer
27 views

STPM CHAPTER 6 DIFFERENTIAL EQUATION [on hold]

dy/d×= y(2×+y)/×(y-×) Given that y=2 when ×= 1
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differential forms and index

The other question i can´t solve is this, If $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$. ...