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

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6answers
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Solution of $(x^2 + y^2)\ dx -2xy\ dy$ = 0

Solve $(x^2 + y^2)dx -2xydy = 0$ The answer is $x^2 - y^2 = Cx$ I've tried the following methods but I'm not getting the answer : Variable Separable (n/a) Homogenous Differential Equation ...
1
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1answer
23 views

A Problem That Involves Differential Equations, Implicit Differentiation, and Tangent Lines of Circles

Here is the Statement of the Problem: Consider the family $\mathbb F$ of circles given by $$ \mathbb F:x^2+(y-c)^2=c^2, c \in \mathbb R. $$ (a) Write down an ODE $y'=F(x,y)$ which defines the ...
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1answer
65 views

Show that $\frac{\int_\Omega|\nabla u|^2+\int_\Omega\alpha|u|^2}{\int_\Omega|u|^2}$ attains a minimum in $W_0^{1,2}(\Omega)$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $H:=W_0^{1,2}(\Omega)$ be the Sobolev space $|\;\cdot\;|_p$ be the seminorm $$|u|_p^p:=\int_\Omega|\nabla u|^p\;d\lambda^n\;\;\;\text{for ...
2
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3answers
287 views

Non linear Differential Equation

Let $\Omega:=\{(x_1,x_2) \subset \mathbb{R}^2 | x_2>0\}$. I want to solve the differential equation $$\begin{pmatrix} \dot{x_1} \\\dot{x_2} \end{pmatrix}=\begin{pmatrix}x_2^2-x_1^2 ...
1
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1answer
15 views

Constructing a linear first order ODE with convergent solutions.

I am studying for a test and cannot figure out for the life of me how to do this problem. I need to construct a first order linear ODE in the form of $y'+p(t)y=g(t)$ such that all of the solutions of ...
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1answer
41 views

Implicit equation. Can it be solved?

Is it possible to find a function $x:[0,T]\to [0,x_0]$ such that, for a fixed $0<\lambda<1$ we have: $$\dfrac{1}{1+\lambda}\left (1-\dfrac{x(t)}{x_0}\right )^{1+\lambda} +\dfrac{1}{1-\lambda} ...
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3answers
40 views

How to solve $y' = -2x -y$

My thought: $\displaystyle\frac{dy}{dx}+x^0y=-2x$ Considering it as the form of linear equation, $\displaystyle\frac{dy}{dx}+P(x)y=Q(x)$ Multiplying $e^{\int1dx} = e^x$ on both sides, ...
1
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2answers
9 views

Find the time that must elapse for the object to reach 98% of its limiting velocity?

I am given the initial value problem $$ \frac{dv}{dt} = 9.8 - (\frac v5) $$ and you are given $v(0) = 0$ I was looking at the solution to this problem. They first solved the differential ...
2
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3answers
56 views

Hints on solving $y'=\frac{y}{3x-y^2}$

$$y'=\frac{y}{3x-y^2}$$ My attempt: $$\frac{dy}{dx}=\frac{y}{3x-y^2}$$ $$dy\cdot(3x-y^2)=dx\cdot y$$ $$dy\cdot3x-dy\cdot y^2=dx\cdot y$$ Any direction? I need hints please ...
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2answers
39 views

Am I solving these initial value problem correctly?

I was just hoping someone could check my work and tell me if I'm solving these types of problems correctly? (Large image version)
2
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0answers
29 views

Advanced calculus: Solving quaternion differential equations

I have a system of two differential equations $$\frac{\partial X(t)}{\partial t}=a_1 A X(t)+a_2X(t) B+a_3 C Y(t)+a_4Y(t) D+a_5$$ $$\frac{\partial Y(t)}{\partial t}=b_1 E X(t)+b_2X(t) F+b_3 G ...
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2answers
42 views

How to show an ODE system has no global solution

Starting from any $(x_0,y_0,z_0)\in \mathbb{C}^3$, can the following ODE system have a solution for all real number? \begin{align} x'(t) &=3 y^2(t) \\ y'(t) &=2 x(t) z(t)-1 \\ z'(t) &=0 ...
3
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1answer
99 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
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0answers
7 views

Usage of Phase Portrait of a system of 2 linear first order ODEs

Let's say have a linear system $\frac{\mathrm{d}\underline{y}}{\mathrm{d}t} = A\cdot \underline{y}$, let say 2 dimensional, and I have $\lambda_1,\lambda_2$ eigenvalues of $A$ and ...
1
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1answer
37 views

Solve of $y''-2y'+y=\frac{3e^t}{1+t^2}+7$

Solve the following DE $y''-2y'+y=\frac{3e^t}{1+t^2}+7$ I can solve for the homogeneous equation, that isn't a problem. However, I don't know how to approach the particular solution. I would try the ...
12
votes
2answers
475 views

Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
0
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1answer
18 views

Solve Sturm-Liouville eigenvalue problem with substitution

I need to solve the SL-eigenvalue problem: $x^4y''+\lambda y = 0$ with $y(1)=y(2)=0$. Therefore one should: 1) substitute with y(x)=xv(x) to get a diferential equation for v(x) and then 2) ...
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2answers
39 views

Solving $\frac{df}{dt}=\frac{i\cdot f}{|f|}$ where $f: \mathbb{R^+} \mapsto \mathbb{C}$

I've never seen a complex DE before, so this is uncharted territory for me. But it's separable so it's easy to turn it into an integral: $$f(t) = \int_0^t\frac{i \cdot f}{|f|} dt$$ Can this be solved? ...
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0answers
19 views

How much of the chemical will be in the pond after a very long time?

A pond initially containing 1000000 gal of water and an unknown amount of undesirable chemical. Water containing 0.01 gram of this chemical per gallon flows into the pond at a rate of 300 gal/hr. The ...
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2answers
16 views

Volume estimation with differential equations

The problem reads: "Using differential equations, estimate the volume necessary to build a tube that is 12m long and has an inner diameter of 25cm and an outer diameter of 25,2 cm." Unfortunately I ...
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0answers
21 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...
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1answer
28 views

Prove that $\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$

Prove that $$\mathcal{L}\left( \int_{0}^t f(u)du \right)=\frac{1}{s}\mathcal{L}(f)$$ I started out with the following identity: $$ \frac{1}{s}\mathcal{L}(f)=\frac{1}{s}\int_{0}^\infty e^{-st}f(t)dt ...
3
votes
1answer
34 views

Estimate for a weak solution to a PDE

Let $f \in L^2(B_R(0))$ and let $u \in W^{1,2}(B_R(0))$ be a weak solution of the equation $$Lu = - \sum_{i,j=1}^{n} D_i(a_{ij}D_ju)+ \sum_{i=1}^{n} b_i D_i u + cu =f.$$ There are constants $0 \le ...
4
votes
4answers
62 views

ODE $2yy'' - 3(y')^2 = 4 y^2$

I'm trying to solve the equation by using these substitutions (how it was suggested in my textbook): $$ y = e^{z(x)} \implies y' = z'y \implies y'' = y((z')^2 + z'') $$ The result is: $$ 2y^2((z')^2 ...
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0answers
18 views

Sturm-Liouville eigenvalue problem of order 4

I want to solve the eigenvalue problem $W''''=\lambda W$ with the boundary conditions $W(0)=W'(0)=W(l)=W'(l)=0$. Has someone a hint how to solve that? Thank you...
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2answers
27 views

How would we know that the particle satisfies both cases?

Consider the differential equation $$\ddot{x}=-n^2 x$$ Now it can be shown that an equivalent formula is $$v^2=n^2(A^2-x^2)$$ , where $A$ is the amplitude of this simple harmonic motion and ...
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0answers
25 views

How Do I solve the Following equation. Getting Confused.

$ (D^4+2D^2+1)y = x^2 cos x $ I applied Inverse Operator case 5 ie $ q(x)= x^m * cos ax $ = Rational Part of $ e^{iax} $ $ 1\over {f(D+ia)} $ $ x^m $ = Rational part of $ e^{iax} $ $ f(D+ia)^{-1} ...
3
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1answer
349 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
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0answers
19 views

The meanings of some symbols in “Calculus of variations”

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked. Thank you very much!!!
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2answers
32 views

Continuous compound word problem using ordinary differential equation

I have a problem with one of my homework questions. (b) A certain bank compounds interest continuously at an annualized interest rate $0<r<1$ (measured in inverse-years), meaning that ...
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1answer
43 views

3rd Euler Equation How to solve

I´m trying to solve this Euler-Cauchy equation $x^{3}y'''+2xy'-2y=x^2ln\left[x+3\right]$ using the video of this link https://www.youtube.com/watch?v=o9Qxq7Bd4aE, but searching here at the site here I ...
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1answer
17 views

Reversing Implicit Differentiation to determine One Parameter Family of Lines

Determine the orthogonal trajectories of the one parameter family of lines y-Cx = 0; Answer is x^2 + y^2 = C Of course you can always do implicit differentiation on each answer from the set of ...
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3answers
19 views

Finding a function whose graph passes through two given points, given its (constant) second derivative

It is known that $y(x)$ passes through the points $(0,2)$ and $(1,4)$. Solve for $y(x)$ if the second derivative is: $$\frac{d^2y}{dx^2} = 1 .$$ The answer is: $$y = \frac{1}{2}(x^2 + ...
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0answers
19 views

What is meant by “homogenous problem” exactly?

Let us look at an entirely linear problem with operator $L$. For an algebraic equation $Lu=0$ is a homogenous equation. If $L$ is a differential operator (PDE or ODE) it has to be supplemented with ...
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0answers
20 views

Is this Riccati ODE solvable? If so, how may I guess the particular solution?

I'm working on a problem and came across this Riccati(?) ODE. Is this solvable? Or must I have two other ODEs for $a(t)$ and $\theta (t)$? $m'(t) = - c_1 \frac{m^2 (t)}{a(t)}\cos(\theta (t) ) - c_2 ...
2
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1answer
48 views

$\int f(y)e^{-y^2} e^{2xy}\,dy$, to prove $f=0$

Show that, if $f \in S(\mathbb{R})$, where $S(\mathbb{R})$ defines Schwartz's space, and $$\int_{-\infty}^{+\infty} f(y)e^{-y^2}e^{2xy} \, dy =0,$$ for all $x \in \mathbb{R}$, then $f=0$ I don't ...
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0answers
17 views

Second order perturbed equation

I've been studying asymptotic behavior on Ordinary Differential Equations. While doing some excercises I found out one excercise which has had me thinking for a while, so I am asking humbly for your ...
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0answers
38 views

A differential equation I

Consider the second order differential equation \begin{align} 2 t^{3} y'' + (5 t^{2} - t) y' + (t^{2} - t + 1) y = 0 \end{align} with the conditions $y(0) = 0$ and $y'(0) = 1$. A solution is known in ...
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1answer
24 views

Use reduction of order to find a solution of the given nonhomogeneous equation.

Question Use reduction of order to find a solution of the given nonhomogeneous equation. The indicated function $y_1(x)$ is a solution of the associated homogeneous equation. Determine a second ...
2
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1answer
23 views

Laplace transform and IVP at $\infty$

Solving the following differential equation $$ty^{''}\left ( t \right )+\left ( t-1 \right )y^{'}\left ( t \right )-y\left ( t \right )=0$$ with initial values $$y\left ( 0 \right )=5, y\left ( ...
0
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1answer
28 views

How to use separation of variables on this differential equation?

Let $a,b,c,d$ be constants. How do I separate $ ay''+b = \frac{c}{(d+y)^3}$ ? I don't need the solution $y=...$, but I need the form $ dy = ... dt$
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1answer
25 views
+100

An applied problem in ODE leading to Interval of Existence issue

ODE books have a section on interval of existence of a solution with a standard set of problems, such as what is the interval of existence for $y'+(\tan t) y = t/(t-2), y(3)=5$, or $y'=y^3, y(0)=2$. I ...
1
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1answer
27 views

Expressing associated Legendre polynomials in terms of unassociated Legendre polynomials

The associate Legendre equation is given as: $(1-x^2)\frac{d^2}{dx^2}y-2x\frac{d}{dx}y+\left[n(n+1)-\frac{m^2}{1-x^2}\right] y=0$ This becomes the standard unassociated Legendre equation for $m=0$. ...
0
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1answer
24 views

Conditions of a differential equation

Consider the differential equation \begin{align} 2 x^2 y'' + x(x^2 - 1) y' + (2 x^2 - x +1)y = 0 \hspace{5mm} y(0) = 0, y'(0)=1. \end{align} A solution readily found is \begin{align} y(x) &= B_{0} ...
3
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1answer
32 views

Is it possible, that the fist two weak eigenvalues of $-\Delta$ in a bounded domain are equal?

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain $\lambda_1$ be the first weak eigenvalue of $-\Delta$ in $\Omega$ $\varphi_1$ be the weak eigenfunction associated with $\lambda_1$ ...
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0answers
37 views

Second Order Differentials: Using $y = A + Bxe^x$

I've went over some of my math work which I'm currently doing at Uni and came across a rather confusing example. The example I went over is based on Second Order Differentials. So basically what I ...
2
votes
1answer
28 views

If $l_i$ is the first weak eigenvalue of $-\Delta$ in a domain $G_i$ and $G_1\subseteq G_2$, then $l_1\ge l_2$ and equality is possible

Let $\Omega_i\subseteq\mathbb{R}^n$ be a domain $\lambda_i$ be the first weak eigenvalue of $-\Delta$ in $\Omega_i$ It's easy to verify that $\Omega_1\subseteq\Omega_2$ implies $\lambda_1\ge ...
1
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5answers
219 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
0
votes
1answer
34 views

Inverting the differential operator $D^2-3D+2$ [on hold]

I am trying to calculate $$(D^2-3D+2)^{-1}(xe^{3x})$$ that is, find a function $f$ such that $(D^2-3D+2)(f)=xe^{3x}$ where $D=\frac{d}{dx}$. Using inverse operator, I am getting an incorrect answer. ...
3
votes
1answer
47 views

Inverse Laplace transform of $\operatorname{arccot}(s)$, $\arctan(s)$

How would one find inverse Laplace transforms of $\operatorname{arccot}(s)$ or of $\arctan(s)$ without knowing in advance that this is related to $\dfrac{\sin x}{x}$?