2
votes
2answers
34 views

Please explain the Quotient Rule

I am currently working on an equation but I'm having a hard time understanding how to get the answer. the answer is ${(x^2-4)(x^2+4)(2x+8)-(x^2+8x-4)(4x^3)\over (x^2-4)^2(x^2+4)^2}$ The equation is ...
0
votes
2answers
15 views

Application in linear differential equations

Hey I'm a little stuck on where to proceed on this question: The initial value problem: $$x′(t) = ae^{−bt}x(t)$$ $$x(0) = x_0$$ arises from a model of tumor growth. The constants a and b are ...
1
vote
1answer
20 views

Linear differential equations and IVP

Hey I'm currently stuck on this question. I don't think its in the form of an integrating factor, but I'm also not sure if its separable? $$\frac {dy}{dx}=-3y+2e^{-x}$$
0
votes
1answer
17 views

Integrating factor with linear differential equations

I've been doing this problem and I'm a little lost on where i am. $$\frac{1}{x} \frac{dy}{dx} - \frac{2y}{x^2} = x\cos(x) ; x>0 $$ So far I think (not sure if right) found integrating factor of: ...
0
votes
4answers
39 views

Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...
0
votes
1answer
25 views

Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why?

Does an inverse Laplace transform for $\hat{F}(s)=e^{-is}$ exist? If not, why? The Bromwich integral is not covered in my course so I can't use it. I'm hoping and guessing that the answer is simple! ...
0
votes
0answers
25 views

Solving Black scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ ...
2
votes
4answers
47 views

Solving first order differential equations

So for this one I'm having trouble isolating for y. If its not possible then in the form with dy with the y variable and x with the x variable. $$\frac{dy}{dx}-2xy=e^{x^{2}}.$$
2
votes
1answer
30 views

What to do when regular approaches fail on linear, non-homogeneous ODES.

In my research problem, I have come across the following form of a time varying, non-homogeneous ordinary differential equation. $$\dot x + \frac{k_1}{t} x = k_2t^{3n}\sin(bt) + k_3 t^n \sin(bt) - ...
0
votes
2answers
32 views

finding values of $x$ in $Z$

Find all values of $x$ such that $\frac{x-4}{2x-3}\in\mathbb Z$? I came up with this question to see if it could be solved based on some other questions I did myself. I thought this could not be ...
1
vote
1answer
33 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
0
votes
1answer
32 views

Please explain the rules of differentiation?

I have an equation $f(x)=3\sqrt{\ x}$ and i have to find the derivative of the function f. What i have gotten so far is $3x^{-1/2}$, which then comes out to be ${3/2}x^{-3/2}$. I know the answer is ...
2
votes
4answers
56 views

Second order homogenous non-linear DE: $3(x')^2 - 2x''x=0$

How do I solve this for $x$? $$3\dot{x}^2-2\ddot{x}x=0$$ $$\Leftrightarrow$$ $$3(x')^2 - 2x''x=0 $$ Note: This comes from my working here(on stack exchange meta sandbox[newest activity]) List of ...
2
votes
1answer
40 views

Solving Nonlinear second order ODE

I want to know how to solve this nonlinear second order ODE. This example is based on the option pricing under the CEV model. $$ \frac{1}{2}\sigma^2(x)u''(x)+\mu u'(x)-Cu(x)=-g(x) $$ where $\mu, C$ ...
1
vote
2answers
95 views

Using logs to find numerical values

If $\frac{log(a)}{log(b)}=1000,$ then what is the numerical value of $\frac{log(a/b)}{log(b)}$? I was not sure on how to solve this. I was just looking at this problem to see how would you solve ...
4
votes
3answers
56 views

Show that the solution of an initial value problem is always less than a given constant

My try is that $$\frac{dy}{dt} =(y-3)e^{\cos ty}$$ $$\frac{dy}{y-3}= e^{\cos ty}dt$$ $$\ln (y-3)=-\frac{e^{\cos ty}}{\sin ty} +c$$ my steps is correct or I made mistakes ? please help to solve ...
0
votes
2answers
26 views

Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
3
votes
2answers
15 views

Differential Equation Word Problem involving y=Ce^(xk) (y=y')

"The rate of change of y is proportional to y. Write and solve the differential equation that models the verbal statement." This part of the problem is easy. My work is such: $y'=ky$ ...
0
votes
2answers
109 views

Find the largest $x$ interval containing $0$ on which $y$ is well-defined.

I'm currently taking an intro course on ordinary differential equations and was given this homework problem: Find the solution of the following differential equation:$$\frac{dy}{dx} = y^2(1-2x)$$ ...
0
votes
0answers
23 views

basic differential equation question

The following statement arises in a proof I am reading, and I do not understand why this is: Suppose $J$ is an open interval containing zero and $x: J \to W$ satisfies $x'(t)=f(x(t))$ and $x(0)=x_0$. ...
0
votes
1answer
30 views

O.D.E of the form $dy/dt=f(y)$

I am working with the following equation: $${\rm d}y/{\rm d}t=y^2(4-y^2)$$ I have thus far done the following: Graphed in an F(y),y-plane found critical points of 0,2,-2 and used first and second ...
0
votes
0answers
20 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
1
vote
0answers
17 views

Doubt on series solution of Legendre's equation

From what I understood: The series solution of an ODE is found using Frobenius Method. For the Legendre's equation: $\displaystyle (1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0 $ The solution ...
0
votes
3answers
36 views

Making a Piecewise Function a Single Function

Is there a way to turn a piecewise function into one function. For example: $$\ f(x)=\begin{cases} g(x) & \text{if $a≤x<b $} \\ h(x) & \text{if $b≤x≤d$} \end{cases}$$ (Can you use the ...
2
votes
3answers
47 views

How to get started with solving basic Differential Equations?

I've just started learning Differential Equations and am having general difficulties with a bit of concepts and on how to actually get started. The problem I have is that the books and sources I find ...
0
votes
2answers
31 views

Formulate the system of 2nd order equations into a system of first-order ODE $y' = f(t,y)$

I have the following system of ODE's $$x'' = -\frac{x(t)}{\left(\sqrt{x^2(t)+y^2(t)}\right)^3}$$ $$y'' = -\frac{y(t)}{\left(\sqrt{x^2(t)+y^2(t)}\right)^3}$$ with conditions and at time $t=0$ we ...
0
votes
0answers
43 views

Is this differential equation linear?

Would an equation like this be considered an ordinary linear differential equation (linear in respect to $y$)? $$\frac{d^3y}{dt^3}\dot{}\frac{d^2x}{dt^2} + \frac{d^2y}{dt^2}\dot{}x^2+\frac{dy}{dt} ...
2
votes
1answer
40 views

Is this an ordinary differential equation?

If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable it is said to be an ordinary differential equation (ODE). If ...
1
vote
2answers
21 views

Solving a differential equation $\int_{0}^x k/x \,\, dx = \int_{0}^t dt$ and $k\,\,dx/dt = x$, where $x=x(t)$ and $x(0) = 0$.

In solving a differential equation $\int_{0}^x k/x\,\, dx = \int_{0}^t dt$ where I tried following: $$\int_{0}^x \frac{k}{x} dx = \int_{0}^t dt$$ $$k[\ln x]_0^x = t$$ where $k$ is constant. But ...
-1
votes
2answers
30 views

Show that a function is a solution to differential equation [duplicate]

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ I know that $\lambda_0$ is a double root in characteristic polynomial. Now I have to show that $y(t) = t e^{\lambda_0 t}$ is ...
1
vote
2answers
57 views

How to prove that a derivative of a formula equals to another formula.

If $u= \ln(\tan x+\tan y+\tan z)$ prove $$\sin 2x \dfrac{du}{dx} + \sin 2y \dfrac{du}{dy} + \sin 2z \dfrac{du}{dz}=2 $$ My answwer was like this: $$u' =\dfrac{ 1}{\tan x+\tan y+\tan z} \cdot( ...
3
votes
2answers
28 views

Question about initial value problem

This seems like it should be something simple. While solving some HW problems I ran across this: $dy/dx = ({1-2x})/y$ which I separated and integrated: $\int{Ydy} = \int{1-2x}dx$ $y^2/2= x-x^2 ...
0
votes
0answers
46 views

Show that a function is solution to differential equation

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ and a function $y(t) = t e^{\lambda_0 t}$. First I am assuming that $\lambda_0$ is a root in the characteristic polynomial. ...
1
vote
1answer
26 views

Show that $ y''+\tan y\cdot(y')^2=0$ when $y=\tan^{-1}(\sin h x)$

I have tried solving it in this manner: $$\tan y=\sin h x$$ Differentiate with respect to $x$: \begin{align*} \sec^2y_1&=h \cos hx\\ sec^2 y\frac{d y}{dx}&=h \cos hx\\ \frac{d ...
0
votes
1answer
58 views

Second order differential equation, orthogonality

A temperature field T(x, t) is determined by the following governing equation: $$\frac 1\alpha\frac {dT}{dt} = \frac {d^2T}{dx^2}$$ (Eq 1) T(x,t) can be expressed as a form of expansion of T(x,t) = ...
1
vote
2answers
41 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
0
votes
1answer
21 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
0
votes
1answer
26 views

Show all solutions of an ODE tend toward a single limit as $t$ approaches infinity and find the limiting value

Given the ODE $2y'+ty=2$ Show that all solutions approach a limit as $t\to \infty$. I've found \begin{align} y=\exp(-t^2/4)\int_0^t \exp(s^2/4) \mathrm{d}s + Ce^{-t^2/4} \end{align} However, I am ...
1
vote
2answers
68 views

Looking for a matrix A(t)

I need your help, I'm looking for a contraexample, I need to give a matrix A(t), such that $$e^{\int_0^tA(s)ds}$$ is not a matrix solution for $x'=A(t)x$. I really don't have any clue what can it be. ...
2
votes
2answers
50 views

Prove second derivative of $g$ is proportional to $g^2$

From Apostol's Calculus Vol. 1, chapter 6.26, exercise 30: Let $f(x) = \int_0^x (1+t^3)^{-1/2} dt$. $a)$ Prove $f$ is strictly monotonic. $b)$ Let $g$ be the inverse of $f$. Show that the ...
0
votes
2answers
35 views

Comparison theorem for ODE

Here is something I'm trying to prove: Conjecture: Suppose $f'(x) \leq \phi(f(x), x)$ and $f(a)=\alpha$. Suppose $g'(x)=\phi(g(x),x)$ and $g(a)\geq \alpha$. Then $f(x)\leq g(x)\,\,\forall x$. ...
1
vote
2answers
32 views

Inverse Trigonometric functions - Boyce & Diprima 2.2.19

The problem asks for a solution to the initial value problem: \begin{align} &\sin(2x)dx+\cos(3y)dy=0\\ &y\left(\frac{\pi}{2}\right)=\frac{\pi}{3} \end{align} The problem is separable and I ...
1
vote
2answers
61 views

Simple Derivative paradox

Suppose I define $y(x)=x^3$ $${dy(x) \over dx} = 3x^2$$ $${dy(x) \over dy} = 1 = 3x^2 \frac{dx}{dy} = 0\text{ since }x \neq f(y)$$ $1 \neq 0$ If you take the differential $d()$ where $dy(x)$ then ...
2
votes
1answer
47 views

solve this ordinary differential equation?

i have the differential equation $y'=\frac{y-x}{y-x+1}$, how i solve this? try: i tryed to substitute $u=y-x$, then $u=y-x\iff y=u+x\Rightarrow y'=u'+1$ then $y'=\frac{y-x}{y-x+1}$ become ...
1
vote
0answers
33 views

Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
0
votes
1answer
25 views

Growth equations

Year 2003 there was approx. 10 % of a substance, the year 2013 the substance is 40 % One modell which can describe the substance speed of growth is that the substance, increases every moment, is ...
0
votes
1answer
40 views

Calculating the constants in the general solution of second order homogeneous ODE reduced from Riccati equation

I am writing a code to simulate a kind of volumetric flow, and I have encountered the non-linear Riccati equation in its general form near the end of my calculations. I am having trouble finding the ...
0
votes
2answers
45 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
0
votes
0answers
19 views

Constant solutions of separable ODE

Consider the IWP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 $$ for continuous functions $g : I \to \mathbb R$ and $h : U \to \mathbb R$ on open intervals $I, U$ with $(x_0, y_0) \in I\times ...
2
votes
1answer
91 views

Differential equation $x\dfrac{dy}{dx}-xy=y^2$

How do I rearrange this ODE so that I can use the basic ODE techniques to solve it i.e. separable, first order using integrating factor and/or Exact ODE. $x\dfrac{dy}{dx}-xy=y^2$