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

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54 views

Hamiltonian function, stationary points

There is Hamiltonian function $H(x,y) = (x - y)^2 + 2sin(x+y)$. I shall show that $(-\pi, -\pi), (\pi, \pi)$ are stationary points of Hamiltionian system and at the same time stationary points of $H$. ...
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4answers
82 views

How to solve the ODE $\ddot{x}=\frac{c}{x}$

How to solve the following ODE? $$\ddot{x}=\frac{c}{x}$$ I have no idea how to solve it since it is not linear. Is there a way of separation or something like that?
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4answers
68 views

Prove $f'(z)=f(z)$ implies $f(z)=ce^z$ for some $c \in \mathbb{C}$ [duplicate]

Suppose that: $$f'(z)=f(z) \text{ for all }z\in\mathbb C.$$ In other words, the complex function $f$ is equal to its own derivative. Prove that there is a constant $c\in\mathbb C$ such that $f(z)=c ...
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87 views

Problem Analysis - Answer but no procedure - Finding Isogonal Trajectories.

I stumbled with this problem in a notebook that has been bothering for the whole day(actually 4)...The answer is written but there's no explanation nor a steb-by-step procedure or anything. If you ...
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1answer
47 views

Is there any method to identify limit cycle in non-linear second order differential equation?

I am working with a second-order non-linear differential equation and I have a very good guess that it should have a stable limit cycle (because of the physics involved in the phenomena which is ...
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64 views

Vector Laplace equation with constraint

I want to solve Laplace equation for a vector $\boldsymbol v=(v_x,v_y)$: $$\nabla^2 \boldsymbol{v}=0$$ but under the constraint that $$(1+v_x)^2+v_y^2=1$$ which becomes $v_y = -(2v_x+v_x^2)^{1/2}$. ...
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2answers
48 views

solution for the following second order linear ODE

Could anybody help me with the solution of: $$a(1-x)^2x^2y''+(1-x)xy'+(x+b)y=0.$$ Consider $a>0$ and $b<0$.
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1answer
47 views

Using integrating factor to find a series solution for an ODE

Consider the ODE $$xy''-y=0$$ Find the solution $y_1$ in the form of a power series, and use an integrating factor in $$y_2=y_1\int\frac{exp(-\int P(x)dx)}{y_1^2}dx$$ to determine $y_2$ To find ...
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1answer
20 views

Linearization of autonomous DE.

I have been working through this PDF about DEs. There was the logistic equation $$\dot{y}=k_0y(1-y/p),$$ where $k_0$ and $p$ are constants. Then a new variable is introduced $y=u+p$. Then ...
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1answer
191 views

Is there an Analytical solution for Blasius equation?

Blasius Equation was introduced to me during my University time ...and I would like to have a solution for it $$y''' + yy'' = 0$$ the $y$ here makes the equation from a linear simple to solve to a ...
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1answer
99 views

Solve the Differential Equation $\frac{dy}{dx}=2+\sqrt{y-2x+3}$

I re-arranged the equation to appear as such: $$1+2x=y+4\cdot\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^2$$ None of the techniques I have learned so far help me to proceed here; particularly, the ...
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2answers
76 views

Finding a continuous function that satisfies a first order differential equation

I'm looking for a continuous function to satisfy the O.D.E.: $$(1+x^2)\frac{dy}{dx}+2xy=f(x);\:f(x)=x\:\:\text{for}\:\: 0\leq x <1;\ f(x)=-x\:\:\text{for} \:\:x>1 ;\ y(0)=0.$$ In my attempt to ...
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1answer
186 views

How to determine the integrating factor for a non-exact differential equation? µ(xy)

$$(3x+\frac6y) + (\frac{x^2}y +\frac{3y}x)y'=0$$ $$ \mu = \mu(xy) $$ I am unsure of how to calculate the integrating factor which depends on x and y. I attempted to solve using $(\mu M)_y = (\mu N)_x ...
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3answers
147 views

Differential Equation of the Form $\frac{dy}{dx}=\sin(x+y)$ [duplicate]

I have been attempting to solve the above differential equation for some time now, and I remain stuck on one step. After substituting $u=x+y$, separating the variables, and integrating both sides, I ...
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1answer
18 views

Solve Differential Equation using Bessel's equation

Is it possible to solve, $$(1-x^2)y''+2(x^2-x-1)y'+(-x^2+2x+7- \frac 1{1-x^2})y=0$$ using Bessel's equation, $$x^2y''+xy'+(x^2-p^2)y=0 ?$$
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1answer
111 views

Solving an IVP with separation of variables

I am trying to solve this equation: $$(e^{2y} - y) \cos(x)\, dy/dx = e^x \sin(x) \text{ with initial condition } y(0) = 0. $$ I already tried to separate the variables, but I couldn't integrate $e^x ...
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1answer
69 views

Population growth equation

How can I answer this problem using the equation $P(t) = P(0)e^{rt}$? Not looking for the math to be done for me, I'm just a little confused with what should be assigned to what variable. Biologists ...
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1answer
60 views

Problem Analysis - Answer but no procedure - Finding Trajectories.

I stumbled with this problem in a notebook that has been bothering for the whole day(actually 3)...The answer is written but there's no explanation nor a steb-by-step procedure or anything. If you ...
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0answers
45 views

Integrating factor (exact equations)

$$(3x + (6/y)) + (x^2/y+ 3y/x)dy/dx=0$$ $$M_y=-6/y^2 $$ $$N_x= 2x/y - 3y/x^2$$ how do I go about finding the integrating factor for this equation? thanks for tips/solutions
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1answer
78 views

General solution to differential equations using Frobenius method

$$x^2y''-2xy'+(x^2+2)y=0$$ The solution for the first indicial root is, $$y_1=a_0\cos x+a_1\sin x$$ and the solution for the second indicial root is, $$y_2=a_0\frac 1x \sin x+2a_1(\frac 1x \cos ...
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3answers
48 views

Problem Analysis - Answer but no procedure

I stumbled with this problem in a notebook that has been bothering for the whole day(actually 3)...The answer is written but there's no explanation nor a steb-by-step procedure or anything. If you ...
1
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3answers
57 views

Basic question about terminology, notation and definitions in calculus

When reading stuff about differential equations I'm coming across some strange (for me) notations/terminology. For example, when coming across something like this: $$\frac{dy}{dt}=f(y,t)$$ or ...
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2answers
88 views

Proving orthogonality of complex form of Fourier Series

I am lost when working on this complex Fourier Series question, I am sure it is a basic simple problem but I am not well versed in applied math: Show that $\{e^{\mathscr i n\pi x/\mathscr l}\}, n ...
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2answers
52 views

1st order ODE, IVP

I am trying to find a solution to: $$y(\ln y)'=\frac{2t}{y}.$$ So far I am unable to start the question. I was given the information that: $$(\ln y)'=\frac{d}{dy}(\ln y).$$
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1answer
39 views

Is it possible to find an explicit form of the solution to $y'=\frac{1-x+y}{x-y}$

We want to solve the differential equation $y'=\frac{1-x+y}{x-y}$, What i did is define $z=x-y$, and then $y'=1-z'$, so overall we have $1-z'=\frac{1-z}{z}$, or in other words $z'=\frac{2z-1}{z}$ ...
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1answer
16 views

Let $\mathcal R$ be an open box. Suppose $\mathcal K$ is compact. Show $\mathcal K \subset \times_{i=1}^n [c_i, d_i] \subset \mathcal R$.

Suppose $\mathcal R \subset \mathbb R^n$ is an open box, that is $\mathcal R = \times_{i=1}^n (a_i, b_i)$ with $-\infty \le a_i < b_i \le \infty$. Suppose $\mathcal K$ is a compact set with ...
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3answers
45 views

Doubts on solving an initial value problem

The problem is : $y(\ln y)'= \frac{2t}{y}$ with $y(0)=0$. How to obtain $y'$ ?
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4answers
86 views

A bullet is shot into the sky, what is its speed when it lands?

During Cinco de Mayo, you shoot a bullet straight up into the sky at the speed of 500 m/s. The altitude of the bullet $y(t)$ at time $t$ seconds after being shot satisfies the differential equation ...
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0answers
59 views

$dy/dx=e^{-x^2} , y(2)=6$

hey so I'm trying to solve this equation using separable method ( an example from my book). It says it isn't an elementary function ( I don't really understand that, but a function that can't be ...
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0answers
148 views

Any online Legendre polynomial calculator?

Do you know if there is any online calculator to calculate Legendre polynomial's coefficients? And combined with a graphing utility would also be desirable. Thank you for your time and help.
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2answers
63 views

Solving differential equation…

There is this differential equation that I could not solve. Can someone please help me solve it? $$y'=-\frac{4t}{y}$$ Thanks in advance!
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1answer
588 views

Using Torricelli's Law

The problem reads: At time t=0 the bottom plug (at the vertex) of a full conical tank of water 16 ft high is removed. After 1 hour, the water in the tank is 9 ft deep. When will the tank be empty? Ok ...
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1answer
145 views

Solution of $\frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0$

I've been trying to to solve the following PDE: \begin{equation} \frac{\partial f}{\partial x}\frac{\partial f}{\partial y} - 2 f \frac{\partial^2 f}{\partial x \, \partial y} = 0 \end{equation} I ...
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1answer
155 views

Picard–Lindelöf theorem

My lecturer says that out of the area D (where differential equation meets the conditions of the Picard–Lindelöf theorem) can't be any solutions for the differential equation, because there, he ...
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1answer
25 views

Help Completing a First Order Separable differential equations

$$ \frac{dy}{dt}=(ty)^2 $$ I am new at this so I just wanted to show my attempt for solving this and see if you guys can tell me where I went wrong? $$ dy/dt=t^2 y^2 $$ then $$ t^2 dt=1/y^2 dy $$ ...
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1answer
89 views

Laplace equation in a rectangle; a non-symmetric solution

Consider Laplace's equation in a rectangle with specified boundary conditions. This problem is solved when $\epsilon_1 = \epsilon_2$ in the following link. $$ \nabla \cdot \epsilon \nabla V=0$$ What ...
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25 views

Did i generalize this series solution correctly?

$$ f(x) = \frac{y''}{p(x)y'} + r(x) y' $$ if all functions are expressed in their power series form, then: $$ y = \sum_{n=0}^\infty a_nx^n $$ $$ p(x) = \sum_{n=0}^\infty p_n x^n $$ $$ r(x) = ...
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2answers
95 views

Is there differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$?

Is there positive differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$? It seems like the answer is no because such a function should have a vertical asymptote ...
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1answer
1k views

Solving a Non-Exact First-Order ODE

Consider the implicit differential equation: $$(45y^3+33xy)dx+(50xy^2+18x^2)dy=0$$ Show that $x^py^q$ is an integrating factor of this equation, and find the explicit values of $p$ and $q$. Then, use ...
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1answer
29 views

Solving first order ordinary differential equation

I was trying to compute the solution for the following differential equation: $$x(2x^2ylog(y)+1)y'=2y$$ As I couldn't get anywhere I checked the hints in the textbook which are the following: ...
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1answer
41 views

Differential Equations - Correct use and placement of constants c1, c2 etc

I am having difficulty trying to discern how to use constants appropriately. For a homework problem $xy^2\frac{dy}{dx}=y^3-x^3$ where $y(1)=2$, I make it all the way (using substitution) to ...
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1answer
91 views

Doubt in Peter Olver “Applications of Lie groups to differential equations”

Book: Applications of Lie groups to differential equations. Second edition (1993). Page: 117-120. Chapter: 2. Section 2.4: Calculation of symmetry groups. Example: 2.41. The heat equation. Question ...
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1answer
118 views

Modelling Concentration

I'm currently doing a research project that involves modelling E. Coli growth in a wetland. The data I've been given is the E. Coli mass concentration ($mgC/L$) at various times throughout the two ...
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0answers
225 views

Uniqueness of solutions to $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$

The problem I am working on is to show that there is a unique compactly supported solution to the PDE $u_{tt} - c^{2}u_{xxxx} + au_{t} = 0$, $(x, t) \in \mathbb{R} \times [0, \infty)$ with $u(x, 0)= ...
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38 views

Problem with an Ordinary Differential Equation Problem

So I have to solve $2t^2y''+(y')^3=2ty'$, I start by making $v=y'$, so then I have $2t^2v'+v^3=2tv$, divide whole equation by $2t^2$, so $v'+v^3/(2t^2)=v/t$, where this is Bernnoulli, so I let ...
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88 views

The method of undetermined coefficients

What's the proof behind the method of undetermined coefficients that's used in solving second order non-homogeneous differential equation with constant coefficients?
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1answer
113 views

Finding the equilibrium solutions of a logistic equation

Given a logistic equation $$dy/dt = r(1 − y/K)y − Ey$$ (a) Show that if $E < r$, then there are two equilibrium points, $y_{1} = 0$ and $y_{2} = K(1 −E/r) > 0$. (b) Show that $y = y_{1}$ is ...
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406 views

Solving a Gompertz equation

(a)Solve the Gompertz equation $$dy/dt = r y \ln(K/y)$$,subject to the initial condition $y(0) = y_{0}$. (b) Data given $[r = 0.71 , K = 80.5 × 10^6 , y_{0}/K = 0.25]$, use the Gompertz model to ...
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1answer
20 views

Finding linearly dependent set of continous real function

Let $C[0,1]$ be set of real valued continous functions on $[0,1]$. Which one of following subsets of $C[0,1]$ is linearly dependent? $\{1, \cos t , \sin t\}$ $\{ \tan^2t, \cos^2t, \sin^2t\}$ $\{1, ...
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1answer
615 views

Volume of Solid Region Between Sphere and Paraboloid

"Find the volume of the solid region above the sphere $x^2+y^2+z^2 = 6$ and below by the paraboloid $z = 4-x^2-y^2$" I am, of course, going to be solving this double integral by converting to polar ...