Solving bernoulli differential equation
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Solving bernoulli differential equation

How to solve $$t \frac{dy}{dt} + y = t^4 y^3$$ First I divided by $t$ to get $$\frac{dy}{dt} + \frac{y}{t} = t^3 y^3$$ Then I multiplied through by $y^{-3...

Published: Aug 12, 2026

Is there really no way to integrate $e^{-x^2}$?
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Is there really no way to integrate $e^{-x^2}$?

Today in my calculus class, we encountered the function $e^{-x^2}$, and I was told that it was not integrable. I was very surprised. Is there really no wa...

Published: Aug 12, 2026

How to find the volume of a tetrahedron?
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How to find the volume of a tetrahedron?

For this question, how do we find the volume of a tetrahedron in $\Bbb R^4$ if we have a 4 by 3 matrix? The cross product and determinant only work for sq...

Published: Aug 12, 2026

Vector dot product = 0 for perpendicular vectors
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Vector dot product = 0 for perpendicular vectors

I'm looking for the intuition behind this property, hopefully one that would be at a high-school level of reasoning. Looking through several of the a...

Published: Aug 12, 2026

Showing an isometry preserves angles
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Showing an isometry preserves angles

If we have two surfaces in $\mathbb{R}^3 ,$ say $S_1, S_2 $. And we have an isometry $f:S_1 \rightarrow S_2$. Then we have two curves $\alpha $ and $\beta...

Published: Aug 12, 2026

When is it allowed to
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When is it allowed to "take apart" a limit (multiplication of limits)?

When is it allowed to "take apart" a limit? Here's an example to show what I mean: $\displaystyle\lim_{n\to\infty}\frac{\frac 1 ne^{\frac 1n}(e-1)}{e...

Published: Aug 12, 2026

What do you get when you take the square root of a negative imaginary number?
updates

What do you get when you take the square root of a negative imaginary number?

Simply, what do I get when I take the square root of negative imaginary number? But, it cant be an imaginary number since the answer to the $2^{\mathrm{nd...

Published: Aug 12, 2026

Prove that $\sin(x)+\cos (x)=\sqrt2\sin\left(x+\frac\pi4\right)$
general

Prove that $\sin(x)+\cos (x)=\sqrt2\sin\left(x+\frac\pi4\right)$

$$\sqrt2\sin\left(x+\frac\pi4\right)=\sqrt2\left(\sin(x)\cos\left(\frac\pi4\right)+\cos(x)\sin\left(\frac\pi4\right)\right)=\sin(x)+\cos(x)$$ Could you so...

Published: Aug 12, 2026

Why we use multiplication rule in an independent event in Probability?
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Why we use multiplication rule in an independent event in Probability?

I know the independent concept and multiplication rules. Example Flip Coin 5 times flip, how many times success of Head? Ans : $\frac{1}{2} \cdot \frac{1}...

Published: Aug 12, 2026

Limit of the ratio of consecutive Fibonacci numbers [duplicate]
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Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. ...

Published: Aug 12, 2026

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