# Tagged Questions

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### 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 ...
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### 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 ...
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### 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? $$\frac {dy}{dx}=-3y+2e^{-x}$$
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### 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: ...
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### 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. ...
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### 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 ...
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### 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) = ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $f(x)\leq g(x)\,\,\forall x$. ...
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### 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 ...
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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 ...
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 ...