Recent content by mathboy20

Homework Statement Let \mathcal{V} \subset \mathbb{R}^n be open and f: \mathcal{V} \rightarrow \mathbb{R}^n be continous. Assume that f has partial derivates which are continous. Then the autonomous differential equation \frac{dx}{dt}(t) = f(x(t)) on the region D = \mathbb{R} \times...

Just looked at the again. There are three possible alphas \alpha = 1/2, \alpha = \sqrt{2} , \alpha = 1 But that doesn't change anything does it?

So anyway if so just to stress that I have understood You correct Mister Tiny Tim. The maximum solution for the original equation lies on the interval between ±tan(απ/2)? Best Regards Mathboy

Initial value problem = IVP But I need to find C in each of the two alpha cases? Don't shoot me, but since its for all t then x oscillates on the interval \pm \rm{t}?? Then as you say alpha < 1. Best Regards Mathboy p.s. If there existed a case where alpha > 1 then the solution...

Homework Statement Given the IVP problem \rm{(1+x^2)^{-1} \frac{dx}{dt} = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)} \iff \rm{(1+x^2)^{-1} dx = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t)} dt Find the maximal solution with the initial conditio condition x(0) = 0. Then alpha is...

Dear Mister Hallsoft, Just to be clear I find the derivative with respect to alpha of the original expression \frac{1}{(x^2+1)} dx = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t) dt \frac{-2x}{x^4 + 2x^2 +1} = \frac{\pi \cdot cos(t)}{4} and then insert the two alpha values to obtain the...

I meant to I take tan on each side of the equation, and thusly obtain x(t) = ?? and then by choosing either alpha1 or alpha2 see which of these gives largest possible solution?

I then get tan(x)^{-1} = \frac{\alpha \cdot 2 \pi}{4} \cdot sin(t) I then take arctan on both sides of equation to arrive at a solution? with regards to the two alpha's?

Homework Statement Dear Friends, Given the differential equation \frac{1}{1+x^2} \frac{dx}{dt} = \frac{\alpha \cdot 2 \pi}{4} \cdot cos(t) with the condition that x(0) = 0 Then find the largest possible solution (this is how its stated) if either \alpha_{1} = 1/2...

Hi I got two tasks which I have some trouble with. 1) A guy has 1770 dollars to shop food for. One bread costs 31 dollars and a jar of jam costs 21 dollars. How many loafs of bread and jar's of jam can the guy buy? I'm suppose to calculate it using Euler Algebra 31x + 21y...

Hi I need help intepreting the following. Given Lotka-Volterra model system \begin{array}{cc} x'_1 = (a-bx_2)x_1 \\ x'_2 = (cx_1 -d) x_2\end{array} Look at the system on the open 1.Quadrant K; where a,b,c,d are all positive constants. Show that the system is integratable, which...

Hi Given z = sin(x + sin(t)) show that \frac{\partial z}{\partial x} \cdot \frac{\partial ^2 x}{\partial x \partial z} = \frac{\partial z}{\partial t} \cdot \frac{\partial ^2 z} {\partial x^2} By using the chain-rule I get: f_x(x,t) = cos(x + sin(1)) f_{xx}(x,t) = -sin(x +...

My solution here is my solution 1. I say B_m "uparrow" if B_m is a subset of B_{m+1} for all m, so they are all nested upward. In this case I want to prove that limsup B_m = liminf B_m = union of B_m over all m. Let B be the union over all m of B_m. Since B_m is a subset of F for...

Let F be the label of an non-empty set and let (B_m)_{m \geq 1} be elements in 2^F Then I need to prove the following: \mathrm{lim}_{m} \ \mathrm{sup} \ B_{m} = \mathrm{lim}_{m} \ \mathrm{inf} \ \mathrm{B_m} = \cup _{m= 1} ^{\infty} B_{m} if B_{m} \uparrow which implies that B_{m}...

Hello and thank Your for Your answer, Anyway if I then have show that the series diverges for all point on the circle of convergens. Doesn't that mean that \frac{2n+3}{2n+1}|z| \geq 1, where n \neq 0 if n = 1 then |z| \geq \frac{3}{5} Am I on the right track here? Best...