0
votes
1answer
12 views

polynomial solution of second order differential equation

Find the polynomial solution $$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$ of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x). Note that this is entry-level calculus, so in my ...
0
votes
0answers
10 views

bibliography for weak solutions of ODE's

Some one could recommend to me some bibliography about weak solutions of ODE's, and solutions of ODE's that are not lipschitz??
0
votes
1answer
30 views

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$

Find $f$ such that $f''(x) = 2+ \cos x$, $f(0) = -1$, $f(\pi/2) = 0$ I integrated it once to get, $2x + \sin x + C$, $C$ being a constant. Then I integrated it a second time to get $x^2 - \cos x ...
2
votes
3answers
79 views

If $f'(x)=f(x)+\int_{0}^{1}f(x)\,dx$ and $f(0) = 1,\,$ then what is the value of $\, \int_0^1 f(x)\,dx=$?

If $\displaystyle f'(x)=f(x)+\int_{0}^{1}f(x)\,dx\,$ and $\,f(0) = 1.$ Then what is value of $\displaystyle \int f(x)\,dx\,?$ $\bf{My\; Try.}$ Let $\displaystyle \int_{0}^{1}f(x)\,dx = A\;,$ Then ...
0
votes
1answer
42 views

Specific form of differential equation

Suppose the function $$ f(x)=p(x)\,\mathrm{e}^{q(x)}, $$ is the solution to a differential equation. From which family of differential equations would $ f $ arise?
1
vote
0answers
13 views

Solving ODE involving matrices

We have a given ODE $ K(x)_{_{3 \times 3}}=xC_1K(x)+x^3C_2K'(x) \tag 1$ where $C_1,C_2$ are constant skew symmetric matrices of dimension $3 \times 3$ with determinant $0$. How do we solve ...
1
vote
2answers
26 views

Find the limit and differential equation

We have the following equality: $$ f(x + \Delta x) = f(x) + a \Delta x \, f(x) - 10 \, b \Delta x $$ with a & b constants. If we take $\lim_{\Delta t \to 0}$ , we get a differential equation. My ...
0
votes
0answers
32 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
0
votes
1answer
15 views

Intermediate Integration Question

I'm having difficulty understanding why $$\int \left[ \left(\frac{dy}{dx}\right) ^2 + \left( y \right) \left( \frac{d^2 y}{dx^2} \right) \right]dx = \left( y \right) \left( \frac{dy}{dx} \right)$$
1
vote
1answer
23 views

Determinant of solution of linear equation

Is there a direct way or method to know if the solution to a linear ODE is invertible? I mean, let $A(t)$ be a ($n$ times $n$) matrix and denote by $X(t)$ an unknown Matrix (of the same dimensions) ...
0
votes
1answer
21 views

integrals and differential equations [on hold]

proof that $x \in \mathbb{R_*^+}$ $\int_{0}^{+\infty} \dfrac{e^{-xt}}{1+t^2}dt=\int_{0}^{+\infty} \dfrac{\sin t}{x+t}dt$ (you can Use :differential equations between two functions)
1
vote
1answer
49 views

Showing that a solution to an ODE is bounded without solving the ODE

Consider the differential equation: $2y'-y^2=-\alpha^2$ where $\alpha>0$ ($\alpha$ is a constant). Ons solution to this equation is $y(x)=\alpha$. Without solving the ODE, show that any bounded ...
0
votes
0answers
8 views

Integration by parts applied to weak form of boundary value proble

In my finite element textbook the proof for strong and weak form equivalence is determined as such: $$\int_0^1w_{,x}u_{,x}dx = \int_0^1wfdx + w(0)h$$ Integrating by parts and making use of the fact ...
0
votes
0answers
41 views
+50

Control Function with solution and fixed initial data on time interval, critical point of a cost functional?

Let $u(t)$ be a solution of the ODE $u''(t)+tu'(t) + u(t) = f(t)$ on the time interval $[0,T]$, with fixed initial data $u(0)=u_0$, $u'(0) = u_1$ where $f(t)$ is a control function. Find $f(T), ...
1
vote
2answers
96 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
0
votes
0answers
26 views

Showing a second order DE has characteristic equation

Verify that $y''-2py'+p^2y=0$ has characteristic equation $(m-p)^2=0$ and has solution $y=e^{px}$ I began by trying to solve $r^2-2p+p^2=0$ but I'm kind of stuck where to go. Any help would be ...
0
votes
0answers
30 views

How do I solve this calculs problem [closed]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
1
vote
3answers
31 views

The limit of a solution of the logistic equation as time tends to infinity

$$ \frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$ What value does $P$ approach as $t$ gets large, ie. as $t \to\infty$. How do I solve this? Is the idea to this question to first rearrange the equation ...
0
votes
0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
1
vote
0answers
19 views

Second Order Differential Equations - Undetermined Coefficients

When solving for this one: $y''-3y'-4y=e^{-x}$ For the trial function, let: $y=Ae^{-x}$ $y'=-Ae^{-x}$ $y''=Ae^{-x}$ $=> Ae^{-x}-3(-Ae^{-x})-4(Ae^{-x})=e^{-x}$ $=> ...
0
votes
1answer
47 views

How does this integration make sense?

I simply don't understand how integration can lead from: $ds^2 = a^2(t) \frac{dr^2}{1 - kr^2}$ to $s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}}$ I appologize, I've never been quite capable of ...
0
votes
0answers
15 views

Existence and uniqueness of SDEs depending on the expected value?

I was thinking of general mean-field SDEs. But let us just look at something really simple: $$dX_t = dt + dB_t, \quad X_0=x$$ the solution to this SDE exists in a strong sense and is: $X_t = x + t ...
2
votes
2answers
398 views

Can anyone explain why this equation using the fundamental theorem of calculus works?

\begin{align} \left| f(b)-f(a)\right|&=\left| \int_a^b \frac{df}{dx} dx\right|\\ \ \\ &\leq\left| \int_a^b \left|\frac{df}{dx}\right|\ dx\right|. \end{align} I do not ...
0
votes
1answer
36 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
1
vote
2answers
35 views

differentiate $y=\sin(xy)$

so I am using chain rule to differentiate this and get down to $ \cos(xy) \times \left( x \frac{dy}{dx} + y \right)$ and then I don't know what to do next. The book says the answer is $\frac{ ...
0
votes
2answers
22 views

Optimization of a rectangular container

A rectangular sheet of tinplate is $2k$ cm by $k$ cm. Four squares, each with sides $x$ cm, are cut from its corners. The remainder is bent into the shape of an open rectangular container. Find the ...
0
votes
2answers
74 views

What is the general solution for $y''e^{-y} =1$? [closed]

how can I find the general solution for an ODE $$y''e^{-y} =1?$$ Thanks.
0
votes
3answers
52 views

Solving the differential equation: $f(x)yy'=(y')^2-0.5$

I am trying to solve this equation: $f(x)yy'=(y')^2-0.5$ I have already tried traditional methods... Any ideas?
0
votes
1answer
25 views

Discuss the following graphs(Differential Equations)

So I have a differential equations midterm coming up soon, and in my last exam I messed the graphing question up. It was very similar to the one I am posting. All the questions said was "Discuss the ...
1
vote
2answers
51 views

When all solutions of $y''+ay'+by=0$ are bounded in R?

Could you please help me solve this problem. Suppose $y''+ay'+by=0$ is differential equation with $a,b$ are real numbers. I need to find conditions when all solutions of this equation are bounded. I ...
1
vote
2answers
38 views

$V dV = \frac{1}{2} d(V^2)$?

I'm following a derivation of the Bernoulli Equation for fluid flows from a book and at one point it says - "Noting that $V dV = \frac{1}{2} d(V^2)$"... How is this derived? Here is what I would do ...
1
vote
3answers
44 views

Integration in question could not be resolved.

I do not know how to solve this integration
1
vote
1answer
22 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
0
votes
3answers
26 views

2nd order odes how do I do it

Why can I just change $y''$ to $m^2$? So for example: $$y''+y'-2y=0$$ $$m^2+ m -2=0$$ $$=(m+2)(m-1)$$ $m=-2,1$ $Ae^{-2x}+Be^x$ But where does that change come from? $y''=m^2$ $y'=2m$ $y=2$ Is ...
1
vote
2answers
63 views

Separation of variables PDEs

In this answer, he has three cases $(\lambda = 0, \lambda \lt 0, \lambda \gt 0)$. I understand the first does imply it is linear, hence it isn't consistent with the initial conditions, and looking at ...
0
votes
1answer
45 views

Flow of a differential equation over what interval

Let $\dot{x}=x^2$. Over what interval is the flow defined? I can see that the solution is of the initial value problem $\dot{x}=x^2$, $x(0)=x_0\ $ is $$ x(t)=\frac{x_0}{1-x_0\cdot t}$$ and that it ...
2
votes
3answers
93 views

Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$

I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation: $$y''+3y'+2y=\exp(\mathrm{e}^x)$$ The homogeneous solutions of this system ...
2
votes
1answer
60 views

Solving the ODE $\,\,x^4yy''+x^4(y')^2+3x^3yy'-1=0$

I'm currently trying to solve the differential equation $$x^4yy''+x^4(y')^2+3x^3yy'-1=0$$ I've tried the substitution $$v=\frac{y}{x}$$which didn't simplify the whole lot. Then I tried rewriting it ...
0
votes
1answer
52 views

Solution of $x^2(y')^2-2(xy-4)y'+y^2=0$

I'm currently trying to solve the differential equation: $$x^2y'^2-2(xy-4)y'+y^2=0,$$ but up to now I've had no succes. I rewrote it as $$(xy'-y)^2+8y'=0$$ and substituted $$v=yx$$ hoping that ...
0
votes
1answer
45 views

Reducing differential equation $\frac{\operatorname d \!y}{\operatorname d \!x} = \frac{(x+y)^2 }{(x+2)(y-2)}$

I'm not able to reduce the following differential equation to variable seperable form. Tried a lot. Please guide.. $$\dfrac{\operatorname d \!y}{\operatorname d \!x} = \dfrac{(x+y)^2 }{(x+2)(y-2)}$$
3
votes
2answers
28 views

Need Help setting up a unusual related rates problem (Calc AB)

Currently I am doing a project in my calculus class where we create a related rate problem relating to 2 ideas pulled out of a hat and solve it(mine was a student(s) bored in class and souls). Being a ...
0
votes
1answer
23 views

Factoring differential equations

I was doing some reading on basic differential equations and the following equation came up: $$ \left(\frac{\text{d}}{\text{d}x} + A(x)\right)\left(\frac{\text{d}}{\text{d}x} + B(x)\right) = ...
0
votes
1answer
30 views

Finding the tangent line to a curve at a given point? Stumped by simple problem.

Obtaining an equation for the tangent of a curve is a problem I've done many times in the past and should be fairly straightforward for simple problems like these. However, I've been graphing my ...
2
votes
1answer
26 views

ODE $y'-xe^y=2e^y$ using $e^{\int P(x)dx}$

I was asked a student how to solve the following problem. Solve for the general solution to the differential equation $y'-xe^y=2e^y$ My first instinct told me that this was a problem that ...
1
vote
1answer
28 views

Second order linear ODE $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$

I have $y^{\prime\prime}+\frac{2y^{\prime}}{x}-\frac{2y}{x^2}=0$ How do I solve this? What have I tried? $1)$ Coupled system: $\begin{pmatrix}y_1^{\prime} \\ ...
3
votes
1answer
38 views

How to solve this linear first order differential equation?

$$\frac{1}{N}\frac{dN}{dt} + 1 = te^{t+2}$$ The equation is separable and so is easily solvable. However doing so gives me the following: $$\int \frac{1}{N}dN = \int(te^{t+2} - 1)dt$$ Simplyifing ...
0
votes
1answer
11 views

Number of solutions for a differential equation?

There are $n$ linearly independent solutions for a $n$th degree differential equation (if the equation has a solution). Could someone explain if the above is true, and if so could someone give me ...
2
votes
1answer
36 views

Differential equation involving a rational function with $\cos(xy)$ and $\sin(xy)$

How to solve this differential equation? $$\frac{ dy}{dx}= \frac{ 3x^2 \cos⁡(xy)-x^3 y\sin(xy)+4x}{x^4 \sin⁡(xy)-8y}$$ The $xy$ inside both sine and cosine are really throwing me off, not even sure ...
1
vote
1answer
20 views

Change of variable in differential equation legitimate?

Just a general question ( I don't want to solve this ODE, I just want to understand why this is legitimate to do or not): Assuming we have the ODE $$y'(x) - \cos(x) y(x)=0$$ on $[0,2\pi]$ Am I ...
0
votes
1answer
28 views

Conditions for the existence and uniqueness of a solution for initial value problems

My book gives the theorem: Let $R$ be a rectangular region in the xy plane defined by $a \le x\le b$, $c \le y \le d$ that contains the point $(x_0, y_0)$ in its interior If $f(x, y)$ and $\dfrac ...