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

learn more… | top users | synonyms (1)

0
votes
1answer
25 views

Approximating solutions for the ODE y'=exp(y/x)

I am currently trying to solve excercise 1-38 from Mathews and Walker. In this excercise I am asked to consider the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\exp(y/x)$$ for two ...
0
votes
2answers
23 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
0
votes
2answers
49 views

Solve for $y' + Py = ae^{bt}$

How do I solve $y' + Py = ae^{bt}$? My attempt: $y' + Py = ae^{bt}\Rightarrow Py - ae^{bt} + 1.\frac{\mathrm{d} y}{\mathrm{d} t}=0$, where $M(t,y)=Py - ae^{bt}$ and $N(t,y)=1$. $M_{y}=P$, and ...
0
votes
1answer
28 views

Solving $y'(t)=\frac{1}{t^2+y^2(t)}$

Solve the following differential equation $$y'(t)=\frac{1}{t^2+y^2(t)}$$ I would appreciate some help with this problem. Thank you very much.
0
votes
0answers
11 views

Solution for an ODE given only at discrete points

The problem I have: For each $n \in \mathbb N$ I have $$\begin{align} x_0^n & \in \mathbb R \\ h_n & \in \mathbb R \\ x_k^n & = x_0^n + k \cdot h_n \text{ for } k \in \{0,1,\ldots n\} \\ ...
2
votes
1answer
33 views

How to solve the ODE $2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$?

I am struggling with this ODE I obtained when solving the Euler-Lagrange equation. Can any one help me with solving the ODE $$2x\frac{dy}{dx}=C(1+(\frac{dy}{dx})^2)^2$$ Thanks so much! It comes ...
1
vote
0answers
10 views

Time period of periodic motion

Find time of one period in polar coordinates $( r, \theta) $ $ \dfrac {d \theta } {dt} = \dfrac{ \sin \psi } {r} $ obeying differential equation in a 2D plane $ \dfrac{d\psi } {dt } = \sin \psi ...
1
vote
1answer
37 views

Solving ODE $x' = \lambda x^2$

I am currently studying continuous dependence ODE theory, and there's one example given in our lecture notes, where I am confused how to solve it. The equation is: $\displaystyle x' = \lambda~x^2$ ...
-1
votes
0answers
31 views

Please guide me what are the topics i need to study in maths from basic. [on hold]

I am not having good knowledge in maths.Please guide me what are the topics i.e (algebra,calculus,diff.eqn...)i need to study by step by step. please guide me.
0
votes
1answer
29 views

Determining the equilibrium solution of a direction field for a first order ODE

Consider the equation $dy/dt = f(y)$ and suppose that $y_{1}$ is a critical point, that is, $f (y_1) = 0$. Show that the constant equilibrium solution $φ(t) = y_1$ is asymptotically stable if $f' ...
1
vote
2answers
51 views

Boundary conditions which yield exactly one solution of the differential equation $u'' + u = 0$

Consider the ordinary differential equation: $u'' + u = 0$. Give an example of boundary conditions which yield exactly one solution $u$. Progress The equation of solutions is $$A\cos x + B\sin x ...
0
votes
0answers
10 views

Let $(I_\eta, y_\eta)$ be maximal with $y_\eta(1) = \eta$ (IVP). Show for $0 < \eta < 1$ we have $y_\eta(t) < t^{\frac 4 3}$, $t \in I_\eta$.

Consider the differential equation $y' = X(t,y)$ with $X(t,y) = \frac 1 3 y^{\frac 1 4} + t^{\frac 1 3}$, defined on $\mathcal D_X = (0,\infty) \times (0,\infty)$. For $\eta > 0$ let $(I_\eta, ...
1
vote
0answers
30 views

What is the process of nondemensionalizing an equation?

Question: I need to scale time by $\frac{1}{I}$ and species by $P$ for the following equation $\frac{dS}{dt}=I(1-\frac{S}{P})-\frac{ES}{P}$ where P - Size of the source pool of species on the ...
1
vote
3answers
281 views

Why do we need $n$ initial conditions to specify the solution of an $n$th order linear DE?

Particular cases show that this is the case. E.g. free fall $\dfrac{d^2 x}{d t^2} = -g$, and the simple harmonic oscillator $\dfrac{d^2 x}{d t^2} + \omega^2x = 0$. I can see why this is so ...
0
votes
0answers
30 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
2
votes
0answers
28 views

A line integral equation popped up when trying to derive Exact ODE integrating factor, can it be solved analytically?

(For convenience, for any functions, only its first instance the x,y dependence will be written out, all subsequent instance the x,y will be suppressed) I have an ODE $$M(x,y)+N(x,y)\frac{dy}{dx}=0$$ ...
0
votes
1answer
14 views

First Order Differential Equation Problem Substitution or bernoulli

I am trying to solve the equation $$dy/dx + xy = y^4$$ using Bernoulli's method but it seems to fail since I end up with $$dv/dx -1/3(xv) = -1/3(v)^-8 $$ I am not sure what to do... Any help would ...
1
vote
0answers
13 views

Time taken to empty a hemispherical shaped tank

The tank has a radius of $2$m when initially filled and has an outlet of cross section $12$ cm2 Outlet flow as I calculated goes according to the law $V(t)=0.6\sqrt{2gh(t)}$. Having found out the ...
1
vote
3answers
32 views

Identify the Differential Equations from the given problem [on hold]

Dear Math expert, Please solve part c of the question. Thanks in advance for your support! I'm able to determine (a) Determine xh and (b) Determine xp. But I'm not able to understand the question ...
0
votes
1answer
20 views

ODE: Why do we change our variable here?

I was trying to solve a matrix equation $\dot x = Ax + Bu$ Rearranging yields $\dot x - Ax = Bu$ Let $I = e^{-At}$ our integrating factor so $d(xe^{-At})/dt = e^{-At}Bu$ Then $xe^{-At}$ = $x_0 ...
2
votes
0answers
21 views

Show that the limit points of a system of differential equations are $p \in D$ and $\partial D$

Consider the following system of differential equations: $ \left\{\begin{matrix} \dot {x}=y-x+x^3\\ \dot{y}=-x \end{matrix}\right. $ By linearization, it's easy to see that $(0,0)$ is a ...
1
vote
1answer
11 views

Why does solving the spherical Bessel equation using Frobenius series produce two quadratic equations for the exponents at the singularity?

The spherical Bessel equation is: $$x^2y'' + 2xy' + (x^2 - \frac{5}{16})y = 0$$ If I seek a Frobenius series solution, I will have: \begin{align*} &\quad y = \sum_{n = 0}^{\infty} ...
19
votes
1answer
3k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
2
votes
0answers
37 views

How to solve $\int_{x}^{x+a} f_X(u) du=e^{-2\lambda_1 x} \int_{x-a}^{x} f_X(u) du$

How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants. Things I ...
0
votes
1answer
38 views

Differential Equation with biology!

I am working on a growth model for bacteria as a function of a nutrient, and I am stuck. So the differential equation I am supposed to be solving is $\frac{dN}{\ DT} = k(C_0 -\alpha N(T)) N$ The ...
4
votes
0answers
61 views

Solution of 2nd order linear ODE with regular singular points, and complex exponents at singularity

The steady state temperature distribution of a rod given by: \begin{equation} \frac{\textrm{d}p(x)y'}{\textrm{d}x} - y = 0,\; 0 \leq x \leq 1,\; \text{and} \;y(0) = 0, \end{equation} ...
0
votes
1answer
22 views

Solution of a Partial Differential Equation

Problem statement Solve $\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}=y$ using the change of variables $\left\{\begin{matrix} u=ax^2+y \\ v=x \end{matrix}\right.$ for a suitable ...
0
votes
1answer
20 views

Can someone verify my derivation of a differential equation involving elliptic integrals, please?

I'm trying to determine the relationship between the major and minor radii ($a$ and $b$, respectively) of an ellipse of constant perimeter and variable eccentricity, and I've been thinking that ...
1
vote
0answers
36 views

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r$ is a smooth real-valued function?

How do I solve the differential equation $r(t)^2 + r^{'}(t)^2 = 1$, where $r: \mathbb R \rightarrow \mathbb R$ is a smooth real-valued function ? In Calculus I've seen linear (higher-order) ...
0
votes
1answer
28 views

If $u : \Bbb R \to \Bbb R$ satisfies $u' + 2\pi x u = 0$, why does $\hat{u}$ (the Fourier transform) also satisfy this?

I'm trying to understand why if a function $u : \Bbb R \to \Bbb R$ satisfies the differential equation $u' + 2\pi x u = 0$, then so does the Fourier transform. The properties I have that I can use ...
1
vote
0answers
17 views

Differential equations. Task. [on hold]

$$f: \mathbb{R}^2 \to \mathbb{R} d_{(x,y)} f =(4x^3y+3x^2y^2)dx + (x^4 + 2x^3y)dy $$ in every point $(x,y) \in \mathbb{R}^2$ Determine: 1) $ \frac{df}{dx}(1,-2)$ 2) $\frac{df}{dh}(2,-3) , h = ...
0
votes
1answer
35 views

Ordinary differential equations of order zero?

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$? If not, can we define differential ...
0
votes
1answer
19 views

Refreshing solving second order ODE

I have a boundary value problem for the following differential equation $$\frac{d^2 v}{d \chi^2} = q^2 \left( v - C \right), \; 0<\chi<S \; and \;\; v(0)=v(S)=0 $$ where $q$ and $C$ are certain ...
0
votes
2answers
32 views

is it possible to intergrate this function to get x(t) and y(t)?

say you have a function as below; $d^2V(t)/dt = -B^2V(t)$ B is a constant Initial conditions $V_x(0) = V$, $V_y(0) = 0$ I can't see how to integrate to get x(t) and y(t); I ended up with ...
2
votes
1answer
26 views

Solution of nonhomogenious differential equations

Kindly help me regarding below math problem. How can I prove? Show that if $y_1(x)$ is a solution of $$y'' + ay' + by = f_1(x)$$ and if $y_2(x)$ is a solution of $$y'' + ay' + by = f_2(x)$$ ...
4
votes
0answers
33 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
0
votes
0answers
17 views

Asymptotic solutions to generalized Airy equation

I am interested in asymtotic solutions, for $x \gg 0$ and $x \ll 0$ of the following differential equation: $\frac{d^ny}{dx^n} + yx = 0$ Here $n$ is an integer $\ge 2$. For the particular case of ...
0
votes
2answers
16 views

Guess maximal solution of ODE ($y^{'} = X(t,y) = \frac 1 3 y^{1/4} + t^{1/3}$) on the form $y(t) = at^p$.

Suppose I have the following ODE: $y^{'} = X(t,y) = \frac 1 3 y^{1/4} + t^{1/3}$ defined on $D_X = (0, \infty) \times (0,\infty)$. I want to guess a maximal solution of the form $y(t) = at^p$ for ...
0
votes
0answers
28 views

A few queries of the method of variation of parameters

I've been reviewing my knowledge on the technique of variation of parameters to solve differential equations and have a couple of queries that I'd like to clear up (particularly for 2nd order ...
2
votes
4answers
110 views

Exponential of matrix

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc.) to calculate this exponential $e^{At}$ with $$A=\begin{pmatrix} ...
1
vote
0answers
32 views

Cauchy-Euler problem [on hold]

I cannot solve this Cauchy-Euler problem. $$x^2y''-xy'+2y=2x$$
2
votes
1answer
41 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
0
votes
1answer
16 views

Solving second order nonlinear ODE given boundary condition at infinity

I am trying to solve the following differential equation $$\frac{d^2 u}{dx^2} = - \frac{d V}{du} \; \; , \;\; where \;\; \; V = \frac{1}{2}u^2 - \frac{1}{4}u^4 $$ And the given boundary conditions are ...
-1
votes
0answers
13 views

Lipschitz continuous function [on hold]

let $y:\mathbb R \rightarrow \mathbb R$ be differentiable and satisfy the ODE $dy/dx=f(y)$ ; $y(0)=y(1)=0$ where $f$ is a lipschitz continuous function then what are the properties of $y$ that it ...
0
votes
0answers
15 views

Determine the equilibrium temp distribution for a 1D rod with the following sources and boundaries.

Q=0 du/dx(0) =0. u(L)=T So ,my attempt is that u(x) = Ax + B, so du/dx = A implies A=0 and so u(L) = 0 + B = T so the solution becomes u(x) = T. But I have a feeling it's not right or I'm ...
0
votes
0answers
11 views

region of xy-plane for which the differential equation unique solution [on hold]

Determine a region of xy-plane for which the differential equation $(y-x)y'=y+x$ would have a unique solution whose graph passes through a point $(x_0,y_0)$ in the region.
0
votes
0answers
22 views

first-order differential equation problem

Given that $y=\sin(x)$ is an expicit function of the first-order differential equation $\frac{dy}{dx}=\sqrt{1-y^2}$. Find an interval I of definition, the solution interval. So I got to the point ...
0
votes
1answer
29 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
3
votes
2answers
149 views

Limit of solution of differential equation without solving the equation.

Given $$x'(t)=A-B\left(x(t)\right)^2, \quad x(0)=0.$$ Is it possible to find $\lim\limits_{t\to\infty}x(t)$ without solving the differential equation? Assuming $\lim\limits_{t\to\infty}x'(t)=0$ gives ...
0
votes
1answer
18 views

Help with an introduction to differential equations?

I am taking linear methods this year and im trying to get some more review for differential equations. This is a problem that I ran across: a) Show that the constant function y(x) = 0, for all x, is ...