0
votes
1answer
34 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
10 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
25 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
31 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
40 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
89 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
16 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
397 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
35 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
21 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
73 views

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

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
62 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
59 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
29 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 ...
0
votes
1answer
20 views

Finding the homogeneous part of general solution to second order non-homogeneous differential equation

I was given this non-homogeneous second order differential equation: $\dfrac{d^2y}{dx^2} + y = sec(x)$ In order to solve this, I need to find the general solution to the homogeneous part by finding ...
0
votes
0answers
24 views

Proof the described sequence obey the formula

The sequence start by solving algebraic equation $ P_1V_1^{1.4}=P_2(V_1-\frac{a}{n})^{1.4} $ for P2 and substitute value of $ P_2$ into next equation $P_2(V_1-\frac{a}{n})=P_3(V_1+\frac{a}{n})$. ...
2
votes
3answers
47 views

Homework Help, Solving particular solution for ODE

(a)Find the particular solution to $y''+2y'+y=\frac{5.5e^{-t}}{t^2+1}$ my question is how to find particular solution when right hand side is negative power (b)$x^2y''+19xy'+81y=x^7$ by Euler's ...
1
vote
1answer
46 views

Find tangent line at given points, no function equation

I have never encountered a problem like this and am a bit confused. Function $f$ satisfies: $f(3)=5$, $f(9)=7$ $f'(3)=11$, $f'(9)=13$ Find an equation for the tangent line to the curve $y=f(x^2)$ ...