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

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3
votes
2answers
93 views

Solving $\frac{d f(x)}{dx} + f(x-1) = x^2$

Given following differential equation: $$\frac{d f(x)}{dx} + f(x-1) = x^2$$ where $ f(x)=0 $ for $x \leq 0 $. How do I find the solution for $ x \geq 0 $ ? I understand that for $ 0 \leq x \leq 1 ...
-3
votes
2answers
30 views

differential equation population problem [on hold]

Consider an initial population of 1000 field mice that grows at a rate proportional to the current population p, so that dp/dt=kp A- What is the IVP the models this scenario? that is, give the ...
-2
votes
1answer
21 views

differntiability of the following function [on hold]

Let f(x)=sinx/x,x≠0 =1 ,x=0, then f is a)discontinuous b)continuous but not differentiable c)differentiable only once d)differentiable more than once
0
votes
1answer
52 views

Doubt on an ODE problem

Consider the following differential equation $$x'(t) = h(x(t))$$ Consider a function $x(t)$ which satisfies the differential equation for $0 \lt t \leq 1$ and another function $y(t)$ for $0.5 \leq ...
1
vote
1answer
42 views

Solution to diff eq

Check whether the function $y=\sin(3x)/3$ is a solution of $xy'+y+3\cos3x$ with the initial condition $y(\pi)=0$ Find $xy'$ for the function $y=\sin(3x)/3$ I am a ex-math minor who is just trying ...
1
vote
1answer
39 views

Solve the equation $3xy''+5y'+3y=0$

For the equation $$3xy''+5y'+3y=0$$ i have to find two independent solutions for $x>0$. And i have to see if this solutions are analytic at 0. My approach: I try to solve this problem using ...
2
votes
0answers
61 views
+50

Uniform continuity of the function $x(t)=e^{tA}x$

Let $A$ be a bounded operator on a Banach space $X$. Consider the exponential function $x(t)=e^{tA}x:=\sum_{n=0}^{+\infty}\dfrac{t^nA^n}{n!}x$, for all $t\in \mathbb{R}$, where $x\in X$. If the ...
0
votes
0answers
23 views

Question on linear partial difference equation with three independent variables $n$, $m$, $k$.

Would it be possible to find closed form for the recursively defined algebraic function of 3 integer arguments F[n,m,k] ? Here are the details: Recursion definition is ...
0
votes
0answers
22 views

Solving system of equations

I have the following set of equations: $y = f(a,b)$ $a = f(y)$ $\dot{b} = f(b,y,\dot{y})$ which I like to solve for $y$. I was wondering if there is some numerical method which I can apply to ...
1
vote
2answers
23 views

What does a number in gradient symbol subscript means?

While solving some problems I have encountered a subscript in front of a gradient symbol. I'm unable to understand it, I know a superscript of 2 on gradient symbol means Laplacian but what does ...
0
votes
0answers
18 views

4th order method

I am asked to solve a ODE using the 4th order Runge-Kutta method, and then given the analytical answer, 'show the method is 4th order numerically' . What does the question 'show the method is 4th ...
0
votes
0answers
16 views

How can I check Lipschitz condition for this function?

I have system of ODE that have these equations: $\partial_t f_1(t,u) = 2 \int_0^1 f_1(x,t) dt + f_1(u,t) \int_0^1 f_2(x,t) dx +f_1(u,t) \int_0^1 f_2(x,t) dx + 8(2u \int_0^1 f_1(x,t) dx - u^2 ...
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 ...
1
vote
1answer
47 views

A basic confusion in the proof of Picard's existence theorem

In the proof of Picard's existence theorem of solution of ODE I don't understand the following step: Once it proves that the limit of uniformly convergent series is a continuous function then it ...
-2
votes
1answer
25 views

Euler's method problem [closed]

Use one iteration of Euler's Method with step size of $h=1$ to approximate the solution to the differential equation at $t=1$: $\begin{array}{l}\frac{{dy}}{{dt}} = {y^2} + t - 1\\y(0) = - ...
-1
votes
1answer
30 views

Is that equation an exact value for differential equation [closed]

Is $2xy^2+4-2(3-x^y)\frac{dy}{dx}=0$ an exact equation? Justify your answer B- solve $2xy^2+4-2(3-x^y)\frac{dy}{dx} = 0$ simplicity.
1
vote
2answers
32 views

Newton's law of cooling problem Differential Equation

Suppose that a building loses heat in accordance with Newton's law of cooling which state the rate of change of temperature within the building is proportional to the difference inn the inside ...
1
vote
2answers
26 views

tank problem Differential equation

A tank is partially filled with 100 gallons of coffee in which 10 lbs of sugar is dissolved. Coffee containing 1/3 lb of sugar per gallon is pumped into the tank at rate 3 gal/min. The yummy ...
0
votes
0answers
26 views

What are the equilibrium solution [closed]

Given $\frac{dy}{dx}= y(y+2)(y-3)^2$ $a-$ what are the equilibrium solution for $\frac{dy}{dx}= y(y+2)(y-3)^2$ $b-$ sketch the phase line and classify all equilibrium pointe $c-$ Next to the phase ...
1
vote
1answer
54 views

Insightful books on differential equations?

What are some recommendations for insightful books on differential equations and difference equations? These books don't need to be in the format of a textbook, and don't need to provide the same ...
2
votes
2answers
56 views

Separation of variables: when to have exponential solution and when sinusoidal?

In separation of variables, one can assume a solution of V(x,y) = X(x)Y(y) and after plugging this into Laplace's equation which is: ${{\partial^2 V} \over {\partial x^2}}$ + ${{\partial^2 V} \over ...
1
vote
1answer
43 views

find matrix A given its exponential

I'm given $e^{At}$ and I need to find A From http://www.math24.net/method-of-matrix-exponential.html I see that $$\frac{d}{dt}(e^{At})=Ae^{At}$$ so does it mean, that to answer my question I just ...
0
votes
2answers
19 views

Particular solution help please

Find the particular solution to the differential equation $y' = \sin x$ given the general solution $y = C - \cos x$ and the initial condition $y(p) = 1.$
0
votes
0answers
45 views

Predator Prey Equation

The Predator-Prey Equation is outlined by the following equation: $$\left\{ \begin{array}{l} \frac{dx}{dt}=\alpha x-\beta xy \\ \frac{dy}{dt}=-\gamma y+\delta xy \end{array} \right.$$ Can someone ...
1
vote
0answers
16 views

Mathieu equation solution with non-periodic boundary conditions

I need to solve the Mathieu equation: $y''(x)+(a-2q \cos(2x)) y(x) = 0$ but with the unusal boundary condition: $y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$ if $\alpha = 0$ than the ...
0
votes
1answer
38 views

Prove uniqueness of differential equation by finding the 'Lipschitz condition'

Refer to Fitzpatrick's Advanced calculus Ex12.3 no. 9. The book tells that If there exist a positive number $M$ such that $|g(x,y_1)-g(x,y_2)|\le M|y_1-y_2|$ for all $(x,y_1)$ and $(x,y_2)$ in the ...
1
vote
0answers
12 views

Finite Difference Discretization of Darcy's law and solving with Picard method

I am trying to discretize Darcy's Law using finite differences and then solving the resulting linear system of equations with the Picard method. So far only in 1D and the steady-state (no time ...
0
votes
2answers
16 views

Problem about moving sides of triangle

Imagine a triangle XOY which sides lie on x-axis and y-axis with hypotenuse XY of length 5 m. Suppose the point X moves away from the (0,0) along x-axis with speed = 1 m per second. What speed the ...
-1
votes
2answers
67 views

Differential equations,x[t] is periodic [closed]

For k > 0,x = x(t),y = y(t) Solve this system: \begin{array}{l} x\frac{{{d^2}x}}{{d{t^2}}} = k\frac{{dy}}{{dt}}\\ x\frac{{{d^2}y}}{{d{t^2}}} = - k\frac{{dx}}{{dt}} \end{array} Picture above is ...
0
votes
2answers
28 views

Equivilent first order differential and initial condition?

I have another homework question that I'm struggling a bit to understand exactly what I'm asked to do. I understand what an initial condition is, but I'm not quite sure how I specify such a condition. ...
1
vote
2answers
40 views

Euler's method for first three approximations?

I have tried variations of the problem for an hour at least and cannot get around to sloving this one. Thank you for input!
2
votes
0answers
41 views

Non-Linear Ordinary Differential Equation in Fluid Dynamics

So while trying to model the physics of a rocket shot from the ground through the atmosphere, I came up with a second-order Non-Linear ODE of the form: $$ \ddot y + \dot y^2 e^y = f(t) $$ This is ...
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
0answers
101 views

Find the function whose Taylor series is $\log(x)+\log(x+1)+\log(x+2)+\ldots$

How do I find a function $f$ whose Taylor series is $$\log(x)+\log(x+1)+\log(x+2)+\ldots$$ for some point $x=a$? It would seem that $$\left.\frac{\partial^n}{\partial x^n}f(x) \right|_{x=a} = ...
1
vote
0answers
57 views

second order ODE :- solution

We have $y''-Py'-Qy = 0 $ where P,Q are $P = K_1+K_2x, Q =K_2 $. $K_1,K_2$ are constants. y' means derivative with respect to x . Please suggest a solution for y. Thanks
-5
votes
1answer
43 views

Differential equation by separation of variables [closed]

Solve the differential equation $dx + e^{5x} dy = 0$ by separation of variables.
-2
votes
4answers
48 views

Proving Differential equation [closed]

Prove that $y=x^3+3$ is a solution of the differential Equation $xy''-2y'= 0$.
0
votes
2answers
41 views

The fundamental difference that determines when a derivative can be calculated directly or only using the chain rule

I was given the following problem: Find $\frac{dy}{dx}$ using the implicit equation $x^2 + y^2 = 1$ What I'm more interested in is the explicit equation, $y = \sqrt{1 - x^2}$ (I'm allowed to ...
0
votes
2answers
21 views

Negative square root solution to Bernoulli equation

Find the solution to the following Bernoulli equation subject to the given boundary condition $$\frac{dy}{dx} + \frac{y}{x} = \frac{3}{2y}$$ $y=-1$ at $x=1$ New dependent variable $z=y^2$. Then ...
1
vote
0answers
12 views

some questions on the soluction of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
3
votes
4answers
740 views

A 6 meter ladder…

A $6$ meter long ladder leans with a vertical wall and top of the ladder is 3 meters above the ground.If it slips at a rate of $2$ m/s then how fast the level is decreasing from the wall? My ...
1
vote
1answer
35 views

Solution to $u_t+\Delta^2u+\Delta u=0$

Suppose there exists a solution to *$u_t+\Delta^2u+\Delta u=0$ of the form $u(x,y,t)=c(t)e^{i\pi(x/4\pi+y/4\pi)}$. I need to find such a function $c(t)$. Plugging $u(x,y,t)$ into *, I got ...
9
votes
2answers
404 views
+150

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
0
votes
2answers
24 views

Finding polynomial satistying potential equation and boundary conditions

Can someone help me with this problem? I know that this polynomial is a solution of Poisson's equation.
1
vote
0answers
20 views

Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d ...
5
votes
2answers
65 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
1
vote
0answers
12 views

Differential Equation Involving a Bivariate PDF and its Marginal CDF

I have a differential equation of the form $$ P(x_1,0) = R(x_1) Q(x_1), $$ where $P$ is an unknown, isotropic bivariate probability density function (pdf) i.e. $$ P(x_1,x_2) = P(x_3,x_4), \quad ...
8
votes
3answers
195 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
2
votes
1answer
84 views

Finding stream function from potential

I have a velocity potential (ie the gradient of this function gives the velocity) given by : $$\phi(x,y) = \frac{1}{\sqrt{(y-1)^2 + x^2}} + \frac{1}{\sqrt{(y+1)^2 + x^2}} + k x$$ where $k$ = ...
0
votes
0answers
12 views

differential equations of the family of curves

Find the differential equation of the family of curves. 1.all the lines tangent to the unit circle $x^2$+ $y^2$ = $1$ 2.all circles through ($1$ , $0$) and ($-1$, $0$)