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

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3
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1answer
37 views

Solving Inhomogeneous Differential Equations Using the Undetermined Coefficients Method

I am trying to solve the following question in my homework: Use the method of undetermined coefficients to solve the following differential equation: $$y'' + y = -\sin(x), \ y(0) = 0, \ y'(0) = ...
0
votes
4answers
104 views

On finding the equilibrium solutions to a system of differential equations

I am asked to find all equilibrium solutions to this system of differential equations: $$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$ and to determine if they are stable, ...
0
votes
1answer
45 views

Flow of a differential equation over what interval

Let $\dot{x}=x^2$. Over what interval is the flow defined? I can see that the solution is of the initial value problem $\dot{x}=x^2$, $x(0)=x_0\ $ is $$ x(t)=\frac{x_0}{1-x_0\cdot t}$$ and that it ...
0
votes
1answer
14 views

DE whose solutions are orthogonal to some vector field

I need to find a differential equation whose solution yields a family of curves in the plane that move orthogonally to the vector field $\langle x-y,y^2\rangle$. Any vector's slope that is orthogonal ...
0
votes
4answers
61 views

How find this ODE $(1-x^2)y''+2xy'-2y=-2$

Question: Find the ODE $$(1-x^2)y''+2xy'-2y=-2$$ I think we can find $$(1-x^2)y''+2xy'-2y=0$$ I have find a solution $y=x$,But I can't find all solution. Thank you
0
votes
1answer
29 views

How do I solve a second order differential equation when y(t) is zero?

I have something of the form: my" + ky' = C My problem here is when I try to take a particular solution and I choose yp = A, the equation becomes 0 = C when plugged in. How do I approach this? Is ...
1
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0answers
18 views

Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
2
votes
3answers
94 views

Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system ...
1
vote
0answers
27 views

Differential equations, derivative of determinant, Euler's formula

Let $b:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth vector field. Let $u(s,x,t):\mathbb{R}^{n+2}\to\mathbb{R}^n$ with $s,t\in\mathbb{R}$ and $x\in\mathbb{R}^n$ satisfy the following differential ...
2
votes
1answer
60 views

Solving the ODE $\,\,x^4yy''+x^4(y')^2+3x^3yy'-1=0$

I'm currently trying to solve the differential equation $$x^4yy''+x^4(y')^2+3x^3yy'-1=0$$ I've tried the substitution $$v=\frac{y}{x}$$which didn't simplify the whole lot. Then I tried rewriting it ...
0
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0answers
16 views

final value theorem in the presence of white noise

I apply the final value theorem to get the steady-state error with the presence of white noise and I just keep getting zero. To me, it seems wrong to have zero steady-state error when there is noise ...
0
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1answer
28 views

Calculus 7th Ed (Stewart) - Chapter 4 solution 2 page 332

This can be really ridiculous for you but I can't understand why dx is up on the root in solution 2 Shouldn't be "du = root(2x+1)*dx" instead of what is show below? Best Regards,
1
vote
0answers
15 views

How do I determine sufficient conditions for the existence of the solution of an initial value problem?

Suppose that $f$ is a smooth function from $\mathbb R^{3}$ to $\mathbb R$ with $f(0,0,0)=0$. Under what sufficient condition will the differential equation $f(x,y,y')$ have a solution satisfying the ...
1
vote
1answer
19 views

derive 2nd order ODE-solution from 1st order ODE

Consider the differential equation $$-\varepsilon v'_\varepsilon + v_\varepsilon^2 = 1\tag{*}$$ for $x\in (-1,1)$ and $v_\varepsilon(0)=0$. Its solution is ...
1
vote
1answer
20 views

Does differential equation always has solution if vector field is only continuous? [duplicate]

Let $V$ be a $continuous$ vector-field defined on a domain $U$ in $R^n$ . Let $x_0$ be a point in $U$. Is it true that the differential equation $\frac{dx}{dt}$ = $V(x)$ admits a solution $\phi(t)$ ...
1
vote
1answer
32 views

Can change of initial condition significantly change the behavior of a non-autonomous system?

For a non-autonomous differential equation $\dot{x}=f(t,x)$, if I change the initial condition from $x(t_1)=x_0$ to $x(t_2)=x_0$, where $t_1 \neq t_2$, will the behavior of the system significantly ...
1
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0answers
57 views

Euler method inequality

Given the problem for $t\neq0$ and $t\le1$ $y'(t)=y^2(t)$ $y(0)=1$ Let $\mu>0$, and $\epsilon_n=\frac12(f(t_{n+1},y_{n+1})-f(t_n,y_n))$, such that $|\epsilon_n|\le\mu|y_n|$ is ...
0
votes
2answers
65 views

Analytical solution to ODE

Does this ODE have a known analytical solution? $$ x(t)''\cdot x(t) = k $$ Here $k$ is a real constant. EDIT: using wolphram alpha, it is clear that it does have an analytical solution. I'm still ...
0
votes
1answer
31 views

If $F''(x) = -\lambda F(x)$ satisfies $F(L)F'(L) \leq F(0)F'(0)$, show that $\lambda \geq 0$.

Consider the function $F \in C^{2}([0,L])$ which satisfies the eigenvalue problem $$F''(x) = -\lambda F(x)$$ and suppose that $F$ satisfies the following constraints on its boundary values ...
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0answers
9 views

Inverse of a sum of cosh, or equivalently $U^{-1}$ for a normal coordinate transformation

I have an equation $\vec{R}_n= \sum\limits_{p=1}^N \vec{X}_p(t) cos(\frac{\pi p}{N+1}(n+\frac{1}{2}))$ of which i know the inverse is given by: $\vec{X}_p= \frac{1}{N+1} \sum\limits_{n=1}^N ...
1
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1answer
28 views

Different general solutions for extremely similar differential equations

Given: $y''' - 5y'' + 3y' + 9y = 0$ and $y''' + 3y'' + 3y' + y = 0$ . Find the general solution of the given high-order differential equations. Starting with the first: 1) $y''' - 5y'' + 3y' + 9y ...
0
votes
1answer
52 views

Solution of $x^2(y')^2-2(xy-4)y'+y^2=0$

I'm currently trying to solve the differential equation: $$x^2y'^2-2(xy-4)y'+y^2=0,$$ but up to now I've had no succes. I rewrote it as $$(xy'-y)^2+8y'=0$$ and substituted $$v=yx$$ hoping that ...
-1
votes
1answer
42 views

Explain scientific error

The third step scientific error, however, the book and the professor says it is 100% correct Please, I want to interpret and explain clearly her and thank you.
0
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1answer
45 views

Reducing differential equation $\frac{\operatorname d \!y}{\operatorname d \!x} = \frac{(x+y)^2 }{(x+2)(y-2)}$

I'm not able to reduce the following differential equation to variable seperable form. Tried a lot. Please guide.. $$\dfrac{\operatorname d \!y}{\operatorname d \!x} = \dfrac{(x+y)^2 }{(x+2)(y-2)}$$
3
votes
2answers
51 views

How to solve $xy'+y+x^4y^4e^x=0$?

I've tried a lot of tricks but I'm still not able to solve the first order differential equation $$ xy'+y+x^4y^4e^x=0 $$ It's not an homogeneous or isobaric equation, it's not a simple linear ...
1
vote
0answers
41 views

Hopf bifurcation how to prove

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=1-(b+1)x+x^2 y\\ \frac{dy}{dt}=bx-x^2 y \end{equation} I now that we will have a bifurcation when $b$ grows and ...
0
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0answers
36 views

a sufficient condition for uniqueness in caratheodory existence theorem.

Let $g(t) , F(r)$ be measurable functions on $(0,a)$ and assume $g(t)$ be nonnegative and $\int_{0}^{a} (g(t)) dt < \infty $ and $F(r)>0$ , and for every $\delta >0$ we have ...
0
votes
1answer
22 views

How to prove $Y_1, Y_2, Y_3$ form a fundamental set of solutions to Linear System $Y' = AY$ when eigenvalues of $A$ are defective

Sorry for the long prose. I am trying to understand a naive treatment of the solution to the Linear System with constant coefficients $$ \left({ \begin{matrix} y_1(t) \\ y_2(t) ...
0
votes
0answers
29 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation by multistep methods?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep methods for the second order ...
0
votes
1answer
59 views

Solve an ODE---usual method no means

How to solve $$(2xy-x^2y-y^3)dx-(x^2+y^2-x^3-xy^2)dy=0.$$ I find that all the elementary method do not solve it. So I turn help from you. Thanks.
3
votes
2answers
28 views

Need Help setting up a unusual related rates problem (Calc AB)

Currently I am doing a project in my calculus class where we create a related rate problem relating to 2 ideas pulled out of a hat and solve it(mine was a student(s) bored in class and souls). Being a ...
1
vote
1answer
41 views

Given one solution of ODE, how to find second solution?

The ODE of my question is $$ (x^2 + 1)y'' - 2y = 2. $$ Solution of homogeneous part of this equation is $1+x^2$. How to find the general solution of this ODE? I did this like, let other linearly ...
0
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1answer
39 views

Solving $\int \frac{1}{x-1}dx$ in two ways.

I have some confusion with this integral $$\int \frac{1}{x-1}dx$$ I can see the solution is $ln(x-1)$ However if I multiply the top and bottom by $-1$ I get $$\int \frac{-1}{1-x}dx$$ And then ...
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votes
0answers
26 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
0
votes
1answer
27 views

limit of solution of a autonomous differential equation at infinitive is a stationary point

$x(t)$ is a solution of $x'(t)=f(x)$ with domain of $(0,\infty)$ for $t$ and assume $x_0 \in \mathbb{R^n}$ and limit of $x(t)$ at infinite is $x_0$. Show that $f(x_0)=0$.
1
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1answer
42 views

global solutions for ODEs

I'm searching for related results for the following problem: Consider the ODE: $$ x'=-3x+ x^2\log x,\quad x(0)=3/2. $$ Does a solution $x(t)$ exist on $t\in[0,\infty)$? A quick research on ...
0
votes
0answers
18 views

Why is the order of the difference operator defined as $p$ rather than $p+1$ for the second order differential equation?

I am reading the book Discrete Variable Methods in Ordinary Differential Equations (1962) by Peter Henrici. I am confused about the accuracy definition in multistep method for the second order ...
0
votes
3answers
31 views

Solve the following Cauchy Euler equation:

How to solve the following Cauchy-Euler equation: $$x^2y''-xy'+y = \ln(x)$$ In class we have solved only homogenous equations, thus I'm not particulary sure how to do this. I tried setting ...
3
votes
1answer
158 views

Frobenius method differ by integer

When the roots of the indicial equation differ by an integer the equation is of the form: $$y_2 (z)=cy_1 (z) \ln(z)+z^{\sigma_2 } \sum_{n=0 }^\infty(b_n z^n )$$ Here is what is bothering me. The last ...
0
votes
1answer
23 views

Factoring differential equations

I was doing some reading on basic differential equations and the following equation came up: $$ \left(\frac{\text{d}}{\text{d}x} + A(x)\right)\left(\frac{\text{d}}{\text{d}x} + B(x)\right) = ...
0
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1answer
39 views

Solving a differential equation with sines and cosines

I'm trying to solve a differential equation which occurs in the proces of computing the following reduced Schrödinger equation by making use of a (Prüfer) transformation $$ y''(x) = (V(x)-1)y(x) \, ...
0
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1answer
31 views

Finding the tangent line to a curve at a given point? Stumped by simple problem.

Obtaining an equation for the tangent of a curve is a problem I've done many times in the past and should be fairly straightforward for simple problems like these. However, I've been graphing my ...
0
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2answers
22 views

How to sketch slope field?

$$y'(x) = (y(x)(1+y(x)))/(1+x)$$ I need to sketch the slope field for this ODE (without using any software) and find the equilibrium solutions. Any help on how to do that is much appreciated.
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2answers
22 views

Uniqueness of IVP solution with a condition weaker than Lipschitz?

We know that Lipschitz condition with respect to $x$ in $$x' = f(t,x) , x(t_0)=x_0 $$ implies uniqueness of IVP problem above. Can we have uniqueness with condition less than Lipschitz?
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0answers
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solution of $y'' =f(x)$ with $y(0)=a$ , $y(1)=b$ by green's function method

Please help me to solve differential equation $y" = f(x)$ with boundary conditions $y(0)=A$ , $y(1)=B$ using Green's Function method.
2
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1answer
26 views

ODE $y'-xe^y=2e^y$ using $e^{\int P(x)dx}$

I was asked a student how to solve the following problem. Solve for the general solution to the differential equation $y'-xe^y=2e^y$ My first instinct told me that this was a problem that ...
1
vote
1answer
30 views

Second order linear ODE $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$

I have $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$ How do I solve this? What have I tried? $1)$ Coupled system: $\begin{pmatrix}y_1^{\prime} \\ ...
0
votes
1answer
47 views

The finite-dimensional function spaces that are closed under taking derivative

I have been stuck on this problem, I don't know where to start. The exact question is: Determine the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$ then ...
2
votes
1answer
31 views

Is the continuity of a vector field enough for the existence of the solution of a differential equation?

I've recently seen the existence-uniqueness theorem for ordinary differential equations from Arnold's book. I understand that the theorem as stated guarantees both existence and uniqueness if the ...
0
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0answers
25 views

Please help me resolve linear different equation

Could you help me to find the solution of linear vector difference equation that given by