Tagged Questions

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

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4
votes
1answer
63 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
1
vote
3answers
46 views

How do I solve this differential equation $y''^2-2y'y''+3=0$ in parametric form?

I do have quite no idea about this one. The obvious substitution $y'=p, y''=p\frac{dp}{dy}$ doesn't make the situation any better
1
vote
0answers
21 views

Application of existence and uniqueness theorem

Let $y_1,y_2$ be solutions on $(a,b)$ to $y'''+P(x)y''+Q(x)y'+R(x)y=0,$ where $P,Q,R$ are continuous functions on $(a,b).$ Let $x_o \in (a,b).$ Suppose $y_1(x_o)=y_2(x_o)=0$ and ...
1
vote
2answers
27 views

Find the first-variational curve which corresponds to the functional $\int_{-1}^1 t^2 \dot{x}^2 dt$ when $x(-1) = -1$ and $x(1) = 1$.

Find the first-variational curve which corresponds to the functional $$\int_{-1}^1 t^2 \dot{x}^2 dt$$ when $x(-1) = -1$ and $x(1) = 1$. Here is what I did: \begin{align} \delta J(x)(h) &= ...
0
votes
1answer
21 views

Linear differential equation and harmonic motion problem

An atom undergoes simple harmonic motion. Initially its displacement is $1$, its velocity is $1$ and acceleration is $-12$ compute its displacement and acceleration when the velocity is square root ...
0
votes
1answer
30 views

Finding $a_n$ from recurrence relation

I am trying to find the closed from for $a_n$ as part of a series solution for an ODE, where the recurrence relation is given by $$a_n=-\frac{a_{n-2}}{n(n-3)}$$ I have come up with ...
0
votes
1answer
35 views

How does one integrate Newton's Cooling Law formula?

How does one integrate $$ \frac{dT}{dt} = k(T - T_0) $$ into this? $$ T(t) = T_0 + Ce^{kt} $$
0
votes
1answer
20 views

Using power series to solve non-homogeneous differential equation?

I've been stuck on this for a while. I've got the following non-homogeneous differential equation and I have to give a solution in the form $\sum_{n=0}^{\infty}c_nX^n$: $$ y'' - 2y'x^2 + 4xy = x^2 + ...
6
votes
2answers
69 views
+50

Explicit traveling wave solution for the diffusion equation

Find explicit formulas for $v$ and $\sigma$ so that $u(x,t)=v(x-\sigma t)$ is a traveling wave solution of the nonlinear diffusion equation $$u_t-u_{xx}=f(u)$$ where $$f(z)=-2z^3+3z^2-z$$ and ...
0
votes
0answers
24 views

natural frequencies of a torsion system [migrated]

Find the angular displacement $\theta_1(t)$ and $\theta_2(t)$ of the system shown in figure below for the initial conditions $\theta_1(0)$, $\theta_2(0)$, and $\dot{\theta}_1(0) = \dot{\theta}_2(0) ...
0
votes
1answer
38 views
1
vote
0answers
28 views

Integration factor of a non exact differential equations [closed]

Find the integrating factor $$(2x^4y^4e^y + 2xy^3 + y)dx + (x^2y^4e^y – x^2y^2 – 3x)dy=0$$
0
votes
1answer
58 views

Clarifying and sketching the differences between proper and improper nodes in phase space for first-order differential equations?

So I recently read this question: Difference between improper node and proper node for phase portrait and I find myself still needing some more concrete clarification about the differences between the ...
3
votes
2answers
56 views

Can one apply a WKB method to an inhomogeneous first order differential equation in order to find the asymptotic expansion of the solution?

Consider \begin{equation} \varepsilon \frac{dy}{dx} = Q(x)y + R(x) \end{equation} where $\varepsilon$ is a small parameter. Can one apply a WKB method to find an asymptotic expansion for the ...
0
votes
1answer
22 views

Calculate rates of change

If a hemispherical bowl has a radius r cm and is binef filled with water at a constant rate. Then how do you show that when the dept of water in the bowl is h cm, then the volume of water in the bowl ...
9
votes
3answers
263 views

How does one show sin(x) is bounded using the power series?

Define the real valued function $$ \sin:\mathbb{R} \rightarrow \mathbb{R}, \qquad given ~~by \qquad \sin(x) := x-\frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots $$ How does one show ...
0
votes
1answer
25 views

General solution of Cauchy -Euler equation

Consider $x^2y''+\frac{5y}{4}=0, \ x>0.$ The general solution is: $c_1x^{1/2}\text{cos}(\text{ln}x)+c_2x^{1/2}\text{sin}(\text{ln}x), \ x>0$ Could anyone advise me how to show ...
1
vote
2answers
65 views

Using laplace transforms to solve a piecewise defined function initial value problem

I want to use laplace transforms to solve the following: $$\frac{d^2 y}{dt^2}+16 y = f(t) = \left\{\begin{array} 1 1&t\lt\pi\\0&t\geq \pi\end{array}\right.\text{ with } y(0)=0 \text{ and } ...
1
vote
1answer
21 views

Non-linear system with all trajectories converging on the line $x=0$, rather than $(2,0)$?

I have the following nonlinear system: $$\begin{pmatrix}\dot{y}_1\\\dot{y}_2\end{pmatrix}=\begin{pmatrix}2y_1\\y_1^2\end{pmatrix}$$ Which I set up to $F=\dot{y}$ Giving the jacobian of ...
1
vote
1answer
25 views

Laplace transform of Bessel's equation

I'm working on what should be a relatively straightforward differential equation. The problem says that the Laplace transform of Bessel's equation leads to (s^2 +1)f'(x) +sf(s)=0. And asks to solve ...
0
votes
1answer
24 views

Solution Using Series Expansion

In order to solve $y'-y=x^2$ using series expansion, do I take the series expansion of $y$ and $y'$ and subtract them and I should get $x^2$?
0
votes
1answer
22 views

Node: Type, Stability, Slope at origin, Trajectories. Linear system.

I have a system of equations: $$\begin{pmatrix}\dot{y}_1\\\dot{y}_2\end{pmatrix}=\begin{pmatrix}2&0\\4&-1\end{pmatrix}\begin{pmatrix}y_1\\y_2\end{pmatrix}$$ Looking at matrix $A$ I can see a ...
2
votes
1answer
35 views

Differential equation (inhomogeneous )

I have been trying to solve this equation for a while. Is there anyone who can help me to solve this ? Any comment appreciated. $$\frac1r \frac{\partial}{\partial r}\left(r\frac{\partial E}{\partial ...
0
votes
1answer
37 views

Separation of variables … $x,y \neq 0$assumption?

When separating variables in a differential equation, often we have to assume that the variables are not equal to $0$ so that division is possible. But what if $0$ is a solution? I'm currently ...
1
vote
1answer
33 views

Integrating DiffEq (and solving for C) yields $C^2=9$ Do you use +3, -3, or both?

Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus from a textbook. I am deeply grateful to the members of this community for their time. ...
2
votes
0answers
49 views

Yet Another Differential Equations Problem

I come from a non mathematical background, so solving differential equations is something that I have to acquire on the go. I hope the following makes sense. I want to chose a nonnegative ...
-1
votes
2answers
37 views

How to solve follow equation? [closed]

I want to solve below equation $$ \begin{cases} \frac{dB}{dt}=-1-aB+\frac{1}{2}bB^2 & \text{} \\ B(T,T)=0 & \text{} \end{cases}$$
0
votes
1answer
33 views

How do you draw a direction field for 2x2 matrix?

I understand $AX = X'$ and by doing so, you get both equation for derivative of $x_1 and $x_2$. When I make a $x_1$ and $x_2$ plot, I am confused regarding which derivative of $x_1$ or $x_2$ to ...
1
vote
2answers
44 views

How to solve a coupled differential equations

I tried different ways to solve this differential equation but I did not succeed. These is the first couple ODEs I try to solve. I hope somebody can give me a hint. \begin{eqnarray} \ddot{x} + ax - ...
1
vote
2answers
40 views

Obtain solution of boundary problem as linear operator.

I'm kinda stuck with a problem right now. I have the boundary problem $$\left\{ \begin{array}{l} -u''(x)+\mu u(x)= f(x), \quad x\in (0,T) \\ u'(0)=u'(T)=0 \end{array} \right.$$ and I have to obtain ...
0
votes
0answers
11 views

Question Related to Boundary Problem

I need to find a formula for the general solution of the initial value problem $$u'' + c^2u = f(x) \wedge u(0) = a \wedge u'(0) = b$$ Then if $0 < c < 1$, I need to find a solution to the ...
1
vote
1answer
49 views

Simple Equation I need help solving [closed]

Let $f(x)$ be a continuous function that satisfies $$\int_0^x f(t)e^{3t} dt = c + x - \cos(x^2)$$ I need to find $f$ and $c$. Please help?
2
votes
2answers
110 views

Fundamental Matrix

Determine $\phi(x,0)$ for $A(x)=\begin{pmatrix} -1 & \cos(x) \\ 0 & -1\end{pmatrix}$, where $\phi(x,0)t_{0}$ is a solution of $\frac{d}{dx}t(x)=A(x)t(x)$. I am not entirely sure as to ...
0
votes
2answers
52 views

Daily life application of differential equations? [closed]

Are there any daily life applications of differential equations ? Do phones or computers use differential equations ? I want applications in daily life not applications at modeling phenomenons in ...
4
votes
1answer
188 views
+50

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
0
votes
2answers
23 views

solve ODE by homogeneous substitution

Solve ode by making substitution. $(x-y) dx + xdy=0$ Let $y = ux$ then $dy = xdu+udx $ $$xdx+(ux-2x)(xdu+udx)=0 \\ xdx+ux^2du+u^2xdx-2x^2du-2uxdx = 0 \\ xdx-2x^2du+ux^2du+u^2xdx-2uxdx=0 \\ ...
1
vote
2answers
29 views

Approach towards second order differential equation

I have the following equation to be solved. Can anybody explain to me how I am supposed to approach this problem? $$4 \frac{d^{2}y}{dx^{2}} + 4 \frac{dy}{dx} + y = (8x^{2} + 6x + 2)e^{-x/2}$$ edited ...
1
vote
1answer
36 views

Solve ${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$

I am wondering how to find an explicit, closed-form solution for the following first-order differential equation: $${y}' = \cosh^{-1}\left ( x \right ) + \mathrm {Si}(x)$$ Where $\mathrm {Si}(x)$ ...
2
votes
0answers
30 views

Fundamental Matrices for Linear ODE

Why is the following statement true?: For a matrix ODE: $\mathbf{x'=Ax}$ with special fundamental matrix, $\Phi (t)$ or $e^{\mathbf{A}}$, where $\Phi(t_0) = I$, and fundamental matrix containing the ...
2
votes
2answers
31 views

Necessary Condition for uniform convergence of series of functions

I would like to make sure that the follwing is a necessary condition for uniform convergence of series of functions: :Let $$ \sum _{n=1}^{\infty }f_{n}(x) $$ be a series of functions, than a ...
1
vote
0answers
13 views

Nondimensionalisation - does it always reduce the number of parameters?

I've just made my ODE system of 8 equations dimensionless, but I've come out with exactly the same number of dimensionless parameters as I had in the system with units. Can nondimensionalisation ...
0
votes
3answers
24 views

Finding homogenous 2nd order differential equation if given general solution

I'm given the general solution to a differential equation: $$y(x)= c_1 \dfrac{\sin(x)}{\sqrt {x}} + c_2 \dfrac{\cos(x)}{\sqrt{x}}$$ This is a uni past paper and I'm absolutely stuck as the homogenous ...
0
votes
1answer
41 views

Strange differential equation (update)

Assume that $f$ is a non-constant analytic function in the variable $x$ such that $$f' = f \cdot P(x,f),$$ where $P$ is a two-variable polynomial over $\mathbf{C}$. Can one have $f(0) = 0$? As noted ...
1
vote
0answers
17 views

Proof that difference equations as asymptotic to their differential analog.

Given a difference equation $a_{n+k}=f(a_n,a_{n+1},\dots,a_{n+k-1})$, we can classify $n=\infty$ as an ordinary, regular singular, or irregular singular point by classifying $x= \infty$ in the ...
0
votes
1answer
15 views

Finding roots of a characteristic equation of higher order ODE

Original DE: $y^{(8)} + 8y^{(4)} + 16 = 0$ The $^{(x)}$ means the order of the derivative. Characteristics equation is: $r^8+8r^4+16=0$ $[r^4 + 4]^2=0$ I don't know how to find the complex roots of ...
3
votes
1answer
62 views

$f(b)-f(a) =((b-a)/2)\cdot(f'(a)+f'(b))-((b-a)³/12)\cdot f'''(c)$ [duplicate]

Let $f$ be three times differentiable on $[a,b]$. $f'''$ is continuous. Show that there is a $c\in[a,b]$ such that $$f(b)-f(a) =((b-a)/2)\cdot(f'(a)+f'(b))-((b-a)³/12)\cdot f'''(c)$$ This looks like ...
2
votes
1answer
66 views

How to prove such a relationship?

I have a function $S(x,y)$ which satisfies the following PDE $$\frac{\partial S(x,y)}{\partial y}=-H\left(x,\frac{\partial S(x,y)}{\partial x}\right)$$ where the known function ...
1
vote
1answer
62 views

Differential equation for Harmonic Motion

Particle undergoes simple harmonic motion. Initially Its displacement is $1$, velocity $1$ and acceleration is $-12$ Compute displacement and acceleration when the velocity is square root of $8$. ...
1
vote
0answers
10 views

Conditional Higher Moments for the Trace of the Wishart Process

I need the second, third, and fourth conditional moment for the trace of the Wishart Process, i.e. $V:= X_{11}+X_{22}$, where the dynamics is given by following $\mathcal{S}^+_2$-valued (non-negative ...
1
vote
1answer
22 views

Finding Eigenvalues for $y''+\lambda y=0$ with boundary conditions.

Given the equation $y''+\lambda y=0$ and boundary conditions $y(1)=0$ and $y(0)+y'(0)=0$. Let $r=\pm\sqrt{-\lambda}.$ If $\lambda >0$ we have ...