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

learn more… | top users | synonyms (1)

-1
votes
2answers
22 views

Second Order Non linear Differential Equation

I have arrived at a differential equation and I need to solve for x. $d^2x \over dE^2$+$Hx$ =$a$($1$+$J\over x^4$ -$1 \over {2x^2}$) Thank you
-4
votes
0answers
25 views

Solutions of the following differential equation

$$\frac{-2q}{k}+z^2+2zp-2zN+(p-N)^2=0$$ What is the solution of this differential equation? Where $N$ is a constant and $p$ and $q$ are the usual notations.
0
votes
1answer
16 views

Connecting a mathematical solution to a differential equation with it's physical solution

I have seen this question in a neuroscience course: It is given after the lecture with these and these slides. I have no background in physics. However, I do know how to solve a differential ...
2
votes
1answer
20 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
0
votes
1answer
7 views

Need help plotting this direction field in Maple: vars must be declared as list [on hold]

I'm having trouble trying to plot this ODE's direction field in Maple. dv/dt=9,8-(v/5) I'm running ...
0
votes
1answer
45 views

Initial conditions for second order ODE with complex stiffness

I'm trying to find initial conditions to ensure systems of the form stay bounded $\ddot{x}_i+\sum_{j=1}^N k_{ij} x_j = 0, \quad k_{ij} \in \mathbb{C}$. For simplicity let's say the $k_{ij}$ lie in ...
2
votes
1answer
37 views

Solving the differential equation $x^2y''+xy'-y=x^2$

$$x^2y''+xy'-y=x^2$$ My attempt: Divided by $x^2$: $$y''+\frac{y'}{x}-\frac{y}{x^2}=1$$ Now to solve the homogenous equation using Euler's method $$y''+\frac{y'}{x}-\frac{y}{x^2}=0$$ To ...
0
votes
1answer
42 views

Python vs Matlab?

I've the problem that I have to transform a function (a rosenbrock-wanner method of 2 order) which is written in python to a matlab-function. Unfortunately I've never done anything with Python and ...
-1
votes
0answers
18 views

Find the greens function of the following non homogeneous problem:

The problem is 100(\left(\fracdy^2dx^2)\right) + y =f(x) with Boundary conditions of y(0)=y'(10pi)=0. \left(\frac12\right) As worked out the general solution is y(x) = Acos(x/10) + Bsin(x/10). ...
0
votes
0answers
19 views

Find two linearly independent solutions of a Legendre equation about $x=0.$

Here is the statement of the problem: Consider the Legendre Equation $$ (*)\qquad (1-x^2)y''-2xy'+2y=0 $$ (a) Find two linearly independent solutions about $x=0$, solving completely any relevant ...
1
vote
3answers
30 views

find the general solution to the following homogeneous differential equation.

$$100\frac{dy^2}{dx^2} + y = 0$$ Is this worked out by using the auxillary equation such that: $$100m^2 + 1 = 0$$ so $m = \pm i\sqrt{1/100}$ ? So the general solution would be $y(x) = A cos ...
0
votes
1answer
22 views

How would you integrate this homogeneous equation?

I am solving a homogeneous equation $\frac{dy}{dx}= \frac{x^2+xy+y^2}{x^2}$ and have come to this step and I'm stuck now with the integration. I could really use some helpful hints to help me $$ ...
0
votes
2answers
14 views

Determining the interval where the solution is valid

I am given the initial value problem $$ y' = \frac{1+3x^2}{3y^2-6y} $$ given y(0)= 1 I have solved this and I got $y^3-3y^2 -x-x^3=-2$. How would I got about finding the interval in which the ...
1
vote
3answers
32 views

function bounded by an exponential has a bounded derivative?

here's the question. I want to be sure of that. Let $v:[0,\infty) \rightarrow \mathbb{R}_+$ a positive function satisfying $$\forall t \ge 0,\qquad v(t)\le kv(0) e^{-c t}$$ for some positive constants ...
2
votes
4answers
80 views

Hints on solving $y''-\frac{x}{x-1}y'+\frac{1}{x-1}y=0$

$$y''-\frac{x}{x-1}y'+\frac{1}{x-1}y=0$$ Is there any simple method to solve this equation? I need hints please $\color{red}{not}$ a full answer
1
vote
0answers
34 views

Prove that a second order diff. eq. has only two linearly independent solutions.

Let $p(t)$ and $q(t)$ be two continuous functions. Prove that the second order linear equation $$y'' + p(t)y' + q(t)y = 0$$ has two, and only two linearly independent solutions.
-2
votes
1answer
26 views

How to prove that the BVP has only the trivial solution? [on hold]

How to prove that the BVP $$x''+f(t)x'+g(t)x=0, t\in[0,1],$$ $$a_1x(0)+b_1x'(0)=0,$$ $$a_2x(1)+b_2x'(1)=0,$$ where $f,g\in C[0,1]$ and $a^2_i+b^2_i>0, i=1,2,$ are constants has only the trivial ...
2
votes
0answers
46 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
7
votes
1answer
67 views

what would a planetary orbit look like if gravity had constant magnitude?

Consider a unit-mass particle that is always experiencing a single unit-magnitude force towards the origin. This is a central force, but it is not one of the familiar ones, e.g. gravity whose ...
0
votes
0answers
30 views

Existence theorems depending on compactness of unit ball?

I can only think of that a semi-continous function attain it's maximum on compact sets. What other existance themorems depend on compactness of unit ball? Which cases are we able to maintain and which ...
0
votes
3answers
28 views

Solving an ODE by inspection

I am trying to solve the following ODE by inspection $$(x-1)y''-xy'+y=0$$ So that method that is recommended is to guess the general form. EDIT : If you guess the general form $y=c_0+c_1x+c_2x^2$. ...
1
vote
1answer
28 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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
3answers
42 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, ...
2
votes
3answers
57 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 ...
0
votes
2answers
40 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)
0
votes
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} ...
2
votes
1answer
55 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 ...
1
vote
0answers
10 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 ...
0
votes
2answers
45 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 ...
0
votes
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 ...
0
votes
2answers
18 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 ...
2
votes
0answers
22 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 ...
1
vote
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
35 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 ...
1
vote
0answers
22 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...
0
votes
1answer
20 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) ...
0
votes
2answers
29 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 ...
1
vote
2answers
43 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? ...
-1
votes
0answers
25 views

How Do I solve the Following equation. Getting Confused. [on hold]

$ (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} ...
2
votes
3answers
289 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
vote
6answers
108 views

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 ...
0
votes
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!!!
0
votes
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 + ...
0
votes
0answers
22 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 ...
1
vote
1answer
25 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 ...
1
vote
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 ...
-1
votes
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 ...