0
votes
1answer
58 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
0
votes
0answers
22 views

Solve the given differential equation by using Green's function method

I am really struggling with the concept and handling of the green's function. I have to solve the given differential equation using Green's function method $\frac{d^{2}y}{dx^{2}}+k^{2}y=\delta ...
3
votes
0answers
59 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
0
votes
1answer
16 views

Using differentials, estimate the difference in the deflection between the point midway on the beam and the point 1 10 ft above it

So I've been trying to figure out the problem for about an hour and I cannot figure it out. Question: To study the effect an earthquake has on a structure, engineers look at the way a beam bends when ...
0
votes
1answer
46 views

Estimate for the limit of the solution of an ODE system

I have this system: $$\begin{cases} \frac{d}{dt}x(t)=-axy\\ \frac{d}{dt}y(t)=axy-by\\ \frac{d}{dt}z(t)=by \end{cases} $$ Let be: $x+y+z=1$ for every $t$ $a>b$ and $a,b$ strictly positive ...
2
votes
1answer
28 views

Dirac Delta Function, Initial Value Problem

Hi I finished this IVP but I cant seem to get the right answer can someone give me some advice as to where I went wrong and point me in the right direction as to how to fix it. Here is the problem and ...
0
votes
0answers
25 views

harmonic oscillator exercise

An object with a mass of 8 kg stretches a spring over 0.06 m This object is drawn further down to 0.30 m and set in motion by an upwardly directed velocity of 0.30m/s in a substance that has a ...
0
votes
0answers
43 views

Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
1
vote
1answer
69 views

Consider the following differential equation$ y'' + 5y' + 4y = 0$.

a) Determine a system of equations $x' = Ax$ that is equivalent to the differential equation. b) Suppose that $y_1, y_2$ form a fundamental set of solutions for the differential equation, and $x(1), ...
0
votes
1answer
31 views

I need help figuring this error percentage homework problem.

Question: Government economists in a certain country have determined that the demand equation for soybeans is given by $p = f(x) = \frac{55}{2x^2 + 1}$ where the unit price p is expressed in dollars ...
0
votes
1answer
43 views

How to go about solving this question on differentials?

A ring of a planet has an inner radius of approximately 52,000 km (measured from the center of the planet) and a radial width of 19 km. Use differentials to estimate the area of the ring. (Round ...
1
vote
2answers
52 views

Modeling salt and water with differential equations

From Differential Equations (Blanchard, Devaney, and Hall, page 35). My question is about the model. I let $x$ be the amount of salt and $t$ the minutes passed. Then ...
4
votes
1answer
83 views

Differential equation $\frac{dy}{dt}=y(1-y)$

I'm starting with $$\frac{dy}{dt}=y(1-y)$$ Then I take the obvious steps. ...
3
votes
1answer
50 views

Every equation of the form $ax''+b(x^2-1)x'+cx=0$, $a,b,c>0$ can be transformed into Van der Pol's equation

Show that every equation of the form $$ax''+b(x^2-1)x'+cx=0,$$ $a,b,c>0$ can be transformed into Van der Pol's equation by a change in the independent variable. I am unable to find this ...
1
vote
0answers
45 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
1
vote
1answer
60 views

Finding functions in a differential equation that satisfies two solutions

"Given a differential equation of the form $a(x)y''+b(x)y'+y=0$, find functions $a(x)$ and $b(x)$ so that $y=x$ and $y=x^2$ are each a solution of this differential equation." I'm really not sure how ...
0
votes
0answers
18 views

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? [duplicate]

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? and in dimension $n$? I know there is singularity when $n=2$.
0
votes
1answer
25 views

Find eigenvalues and eigenfunctions of this BVP: transforming the eqn?

so I've been stuck on this problem after my first attempt. I got the trivial solution after using the characteristic equation $3r^2 - 4r + 1 = 0$... Find all eigenvalues and associated eigenfunctions ...
2
votes
0answers
15 views

If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit. [closed]

Let $X$ be a field in $\mathbb{R}^3$, $C^1$ class. If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit.
1
vote
0answers
17 views

Maximum intervals of a solution and singularities [closed]

Let $X$ be a vector field of $C^1$ calsse in $\Delta \subseteq \mathbb{R}^n$. Prove that if $\varphi(t)$ is a trajectory of $X$ defined maximum range $(\omega_-,\omega_+)$ with: $$\lim_{t \rightarrow ...
2
votes
0answers
20 views

For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
3
votes
1answer
110 views

Derivative of the parameter

I have the equation$\begin{cases} x'(t)=x(t)+y(t) \\y'(t)= \mu y^2(t)+x(t)\end{cases}$ Cauchy problem $\begin{cases} x(0)= 1 + \mu \\y(0)=-2\end{cases}$ . I must calculate $\frac{\partial ...
4
votes
5answers
257 views

Shorter solution to differential equation?

I'm looking for a shorter way to find a maximal solution to the differential equation $$y''-2y'+y=xe^x+e^x\cos(x)$$ $$y(0)=y'(0)=1$$ At first I was hoping I could convert the right side to ...
1
vote
1answer
55 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
1
vote
2answers
40 views

suppose that $(\frac{2x}{y^3}-3x^2)dx+\frac{3(1-x^2)}{y^4}dy=0$ What is $y=?$

I have a question: suppose that $$(\frac{2x}{y^3}-3x^2)dx+\frac{3(1-x^2)}{y^4}dy=0$$ What is $y=?$ Thanks ahead:)
0
votes
0answers
37 views

Green function of Sturm liouville problem

How to find the Green function of the following problem: $$\begin{cases}-(p(t)u')'+q(t)u=f(t,u), t>0\\u(0)=u(+\infty)=0\end{cases}$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in ...
0
votes
2answers
42 views

Solve the boundary value problem (C.S.I.R)

Consider the boundary value problem $$ -u''(x) = \pi^2 u(x) , \ \ x \in (0,1) \ \ and $$ $$ u(0) = u(1) = 0$$ if $u \ \ and \ \ u'$ are continous on $[0,1]$, then $\int_0^1 u^3(x) dx = 0$ ...
2
votes
0answers
52 views

General solution of ODE

please what is the general solution of $$-(p(t)u')'+q(t)u=0$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in L^1((0,+\infty))$ Thank you
1
vote
1answer
23 views

Solving $x' = Ax$ for real $x$ where $A$ is a matrix with complex eigen values

I have the following linear differential equation system: $$x' = A x$$ where $$ A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & -2 \\ 2 & 2 & 1 \end{array} \right) $$ I ...
0
votes
1answer
41 views

characteristic equation of differential equation

Given $x''+3x'+2x=4.$ ($''=2nd $ derivative, $'=1st$ derivative) Determine the characteristic equation of this differential equation. I'm having a hard time doing this because of that $4$. Any help ...
1
vote
0answers
18 views

Simple differential equation modelling question.

The question is: A chemical dissolves in water at a rate equal to 10% of the amount of undissolved chemical per hour. At time $t$ hours the amount of undissolved chemicalis $x$ grams. Initially the ...
0
votes
1answer
30 views

Fastest way to compute minimal polynomial (for solving $x' = A x$, $A$ matrix)

In general, given a $3\times 3$ or $4\times 4$ matrix $A$ which doesn't have a lot of $0$ entries, what is the fastest or less error prone way to compute its minimal polynomial? More generally, I ...
0
votes
1answer
23 views

Stability of solution ODE with parameters

For which $a,b \in \mathbb{R}$ the solution $x=y=0$ is stability for system of ODE: $x'=ax+y+x^2$ $y'-x^2+by+y^2$ In some cases it's easy, because we can ...
0
votes
1answer
25 views

Classify stationary points

I came up with two stationary points in a system of three differential equations. To classify them, I calculated the eigenvalues of the Jacobian, evaluated in this stationary points. For the first ...
0
votes
0answers
26 views

problem on partial integration in two variables

I am trying to find the solution to an assignment, the question is, to find the solution to the following fourth order differential equation in two variables; $D^4w/Dx^4+2D^4w/Dx^2Dy^2+D^4w/Dy^4=q/k$ ...
1
vote
0answers
22 views

How to I find $y(t)$ when the flow rate in is $=r$ and concentration of chemical $Y$ coming in is $=X$ grams per liter?

A tank initially contains $1$ liter of water and $628$ grams of chemical $Y$. A solution containing $X$ grams per liter of chemical $Y$ flows into the tank at the rate of r liter/hour. The mixture ...
1
vote
0answers
25 views

Poisson differential equation

I'm stuck on an old exam question: Let $\Omega = \{(x,y) \in R^2 : 1 < x^2 + y^2 < a \}$. Determine the unique solution for the following boundary condition problem: $\Delta u = 1$ for $(x,y) ...
0
votes
2answers
43 views

Solve the System of Linear Differential equation $\frac{dy}{dt} = Ay$

Consider $A = \begin{pmatrix} 0&1&0\\0&0&1\\0&0&0 \end{pmatrix}$ and y= $\begin{pmatrix} y_1(t)\\y_2(t)\\y_3(t) \end{pmatrix}$ satisfy $\frac{dy}{dt} = Ay$ ; t>0 ; $y(0) = ...
2
votes
1answer
64 views

Hint for solving $ y (y')^2 + (x-y) y' - x = 0$

Need to solve the following ODE: $$ y (y')^2 + (x-y) y' - x = 0$$ I don't really know how to start. Any hints?
3
votes
2answers
26 views

Solve $ 3 e^x \tan{y} \, dx + \dfrac{2-e^x}{\cos^2{y}} \, dy = 0 $ Stupid error somewhere

I am trying to solve the following ODE $$ 3 e^x \tan{y} \, dx + \dfrac{2-e^x}{\cos^2{y}} \, dy = 0 $$ This is my attempt: Its form looks like, $$P(x,y) \, dx + Q(x,y) \, dy = 0$$ so I may be exact ...
3
votes
1answer
29 views

Solve $y^{2/3}+(y')^{2/3}=1$ other than the direct method?

Is there any way to solve $$y^{2/3}+(y')^{2/3}=1$$ other than just solving for $y'$ and then integrate?
2
votes
3answers
85 views

Solve $y' = \frac{1}{x\cos(y) + \sin(2y)}$

I need to solve this ODE $$ y' = \dfrac{1}{x\cos(y) + \sin(2y)}$$ Could you give me any hints? I don't even know how to start.
0
votes
1answer
33 views

Tangent to integral curve

I have an equation like: $$4y'=y(x^2-4x-3)$$ and I have to find the equation to the tangent to the integral curve, which goes through a random point from the square $K =$ $\{-5≤x≤6,-6≤y≤5 \}$. I am ...
1
vote
2answers
36 views

How to solve this ODE (integration factor?)

Im trying to solve the following ODE: $(x+y+1) dx + (2x +2y -1) dy = 0$ In the theory of my book these presented with the form $P(x,y) dx + Q(x,y) dy = 0$ So for my example we have $P(x,y) = x +y ...
0
votes
0answers
24 views

Correct me if I am wrong (homogeneus ODE)

Solve the following ODE: $x y' = \sqrt{x^2 - y^2} +y $ This is my attemp: $y' = \dfrac{\sqrt{x^2 -y^2} +y}{x}$ Now $z = \dfrac{y}{x}$ thus $y' = z'x + z$ and using this change $z' x = \sqrt{1 - ...
1
vote
1answer
35 views

Help required to solve an ODE

To Solve: $\displaystyle \left [1+\left(\frac{dy}{dx} \right)^2\right]^{3/2}=a\frac{d^2y}{dx^2}$ My Attempt: Take $\displaystyle \frac{dy}{dx}=p$ Now we have: $\displaystyle \left ...
4
votes
1answer
38 views

To Solve an ODE

To Solve: $\displaystyle (1+x^2)\frac{d^2y}{dx^2}+1+\left(\frac{dy}{dx}\right)^2=0$ My Attempt: Take $\displaystyle \frac{dy}{dx}=p$ Now we have: $\displaystyle (1+x^2)\frac{dp}{dx}+1+p^2=0$ ...
0
votes
2answers
34 views

Simultaneous Total Differential Equations 2

To Solve : $\displaystyle \frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz} $ Any hints?
1
vote
0answers
22 views

Simultaneous Total Differential Equations

To Solve : $\displaystyle \frac{dx}{x^2-yz}=\frac{dy}{y^2-zx}=\frac{dz}{z^2-xy} $ This type of problem is usually solved using a) method of grouping (doesn't seem useful here as any of x, y and z ...
0
votes
1answer
24 views

Notation confusing my understanding of a homework problem

Probably ultra simple, but asking google about notation is non-trivial in a case like this. The text is Oksendal's Stochastic Diff Eq and, very simply, the question is as follows: Let $B_t$ ...