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

learn more… | top users | synonyms (1)

0
votes
0answers
10 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.
-1
votes
1answer
20 views

How to prove that the BVP has only the trivial solution?

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
38 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
39 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
22 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
27 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
25 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
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 ...
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
0answers
33 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
8 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
43 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
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 ...
2
votes
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 ...
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
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 ...
1
vote
0answers
20 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
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) ...
0
votes
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 ...
1
vote
2answers
40 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.

$ (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
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
vote
6answers
106 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
20 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
24 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 ...
1
vote
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 ...
1
vote
1answer
68 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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$
0
votes
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 ...
3
votes
1answer
34 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 $\lambda_1$ ...
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
vote
1answer
30 views

First ODE problem solution different than WolframAlpha solution

$-y'' +2y' - y = x$ , with conditions $y(0) = y(1) = 0$ I am supposed to find a solution for this problem, so I started with finding the result for the homogeneous equation, and i got $y = c_{1}e^x + ...
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. ...
0
votes
1answer
25 views

How do I find the Laplace Transform of $ \delta(t-2\pi)\cos(t) $?

How do I find the Laplace Transform of $$ \delta(t-2\pi)\cos(t) $$ where $\delta(t) $ is the Dirac Delta Function. I know that it boils down to the following integral $$ \int_{0}^\infty ...
0
votes
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} ...
0
votes
0answers
21 views

clarity in the solution of the following problem

$$(D^2+D)y=x^2+2x+4$$ I found the solution as $$CF=C_{1}+e^{-x}C_{2}$$ and PI=$$\left(\frac{x^3}{3}\right)+4x$$ but the solution from my teacher is PI = $$\left(\frac{x^3}{3}\right)+4x+C3$$ Where ...
1
vote
5answers
220 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*} ...
2
votes
0answers
22 views

Phase line and Equilibrium Points

Consider the differential equation $dy/dt=y^8+3y^6-y^2-1$. Sketch the phase line and classify the equilibrium points. Since when $y=0$, the derivative is negative and when $y>1$ the derivative ...