Recent content by LAZYANGEL

You have ux=sin(x)*u Can't you rewrite it as: y'=y*sin(x)? Solve it as ODE and instead of writing the constant, just assume it's a function of the other variables.

Infinity and the infinitesimal cannot be treated as straight up values for arithmetic operations, infinity is not a number you can reach. Defined by the limit as values for x gets bigger and bigger in contrast with constants in the expression. Just like what DrClaude said.

Use these values for the right hand side: ##i^{\frac{1}{4}}=\left ( e^{i \frac{\pi}{2}}\right )^{\frac{1}{4}}=e^{i \frac{\pi}{8}}=\cos{\left (\frac{\pi}{8}\right)}+i \sin{\left (\frac{\pi}{8} \right )}##

Hey folks, found a couple of interesting integrals and was able to solve one of them ANALYTICALLY! That means no numerical solutions needed. $$\int_{0}^{\frac{\pi}{2}} \frac{1}{1+(tan(x))^{\sqrt{2}}} dx$$ $$\int \frac{1}{1+e^{\frac{1}{x}}} dx$$ The first one I solved and will reveal analytic...

I've modified it a bit: ##\forall x \: [C(x) \rightarrow F(x)],\: x \in \mathbb{Z}## All comedians are funny. ##\exists x \: [C(x) \rightarrow F(x)],\: x \in \mathbb{R}## Yet only some of them really are.

First of all, this is NOT a homework question. So I give you three barrels, one has 50 apples, the second has 50 oranges, the third has 50 - a mixture of both apples and oranges. However, I have intentionally placed the wrong labels on all of them. Find the minimum number of fruits you need...

For the quantum mechanics in your chemistry class for high school you'll barely need any calculus. For basic quantum mechanics (college level) you will need up to multivariable calculus, some understanding of differential equations (little bit about PDEs), linear algebra, and understanding of...

Trial and error is the way to go, unless you want to put the log product function into play.

I'd use comparison test with another series that has similar rate of change as the one you're using. ##\sum \limits_{i} \left | {b_i} \right | \geq \sum \limits_{i} \left | {a_i} \right |## Where ##a_i \approx b_i## in structure, but ##a_i## is both monotonic and bounded.

##m_e=\frac{\int \limits_{0}^h m(z)\phi^2(z)\:dz}{\int \limits_{0}^h\phi^2(z)\:dz}## This formula looks a lot like the centroid equation: ##\bar{x}=\frac{\int x\:dA}{\int dA}## And for that there is a theorem called the Pappus theorem, and it says that if you rotate the area around a...

Yes but since you've redefined the function as ##x^{\frac{2}{2}}## and applied the chain rule, you've essentially taken the derivative of a fractional exponent. You can say that ##\forall \: x^n## the derivative has a singularity at zero where ##n \in (0,1)##.

##\lim\limits_{x^2+y^2\to\infty} x y e^{-(x+y)^2}## ##\lim\limits_{r^2\to\infty} r^2 \cos(\theta) \sin(\theta) e^{-r^2 (1+\sin(2\theta))}## ##\lim\limits_{r^2\to\infty} \frac {r^2 \cos(\theta) \sin(\theta)} {e^{r^2 (1+\sin(2\theta))}}## The limit is undefined when the ##e^{r^2 (1+\sin...

Take a simple definite integral like ##f(x)=x## and use simple limits. ##\displaystyle\large\int_0^x f(x) \: dx=\lim\limits_{n\to\infty} \sum\limits_{i=1}^n f(x^*_i) \Delta x## ##\displaystyle\large\int_0^x x \: dx=\lim\limits_{n\to\infty} \sum\limits_{i=1}^n (x_i) \frac{x}{n}##...

Very simple, define ##g(x)=x^2## and ##f(x)=\sqrt{x}## Take the derivative of ##f(g(x))## ##\frac{d}{dx} \left [ f(g(x)) \right ] = f'(g(x)) \cdot g'(x) = \frac {1}{2 \sqrt{x^2}} \cdot 2x = \frac{x}{\sqrt{x^2}} = \frac{x}{\left | x \right |} = \mathrm{sgn(x)}##

Integrating factor sounds way easier, because the integrating factor eliminates the exponential on the right hand side which is fantastic. ##\mu (x)=e^{\int 0.025 \: d\theta}=e^{0.025 \theta}## And when you multiply both sides by the integrating factor, you get: ## \frac{d}{d \theta} \left [...