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## PDF Editor FAQ

## What are some great business ideas that you wouldn't mind giving away for free?

Here are some stats:Americans spend $48B a year on plants, but there isn’t a single DTC retailer that does more than $10m a year. That’s a lot of space to capitalize on.The sneaker resale market is estimated to be worth $2B, and is expected to triple by 2025. There’s lot of money to be made, and it’s not too late to jump in.Nearly 300m tons of plastic is produced annually, of which an estimated 50% is single-use. Bali recently banned single-use plastics, and new solutions will likely emerge worldwide. Companies that produce petroleum-free, single-use alternatives (like cassava starch) will be massive.All of these ideas are from Trends’ weekly newsletter. Our team of analysts pore over data, find the trends that matter, and deliver the insights you need to understand where business is going and how to capitalize.As part of the Trends’ community, you have access to our forum of likeminded entrepreneurs and founders so that you can ask questions, connect, and collaborate on ideas.The opportunities are endless if you know where the white space is.

## With Uber’s valuation now down -20% due to SoftBank’s $48 billion tender, what are some reasons why employees stay despite earning below market salaries?

There are a few incorrect assumptions in your question:Firstly, SoftBank invested $1.25B at the same $68B valuation as the previous round. So this is a flat round, not a down round. While a flat round isn’t great result, I am guessing a few factors contributed to the lower valuation.So why was it a flat round?409A valuation importance increasing as company reaches scale. I have to be careful about what I say here due to confidentiality, but our 409A valuation, which is conducted by a third party, may have played a part in the SoftBank deal. At most startups, the 409A valuation lags behind VC valuations. But as a company matures, the gap narrows, as there is simply less growth potential for larger companies.Uber sacrificed valuation for other deal benefits. As far as I can tell, the SoftBank deal is a very good thing for Uber. So I think our board was probably willing to allow SoftBank to negotiate heavily on the valuation.This deal will reduce Board tension by taking voting shares away from Travis and allowing investors to pay out their LP’s.I believe it also allowed some early employees “trapped” at Uber by their stock options, to sell and finally move on to another job.Ok, so what about the $48B valuation? Well, the $48B valuation is for a different type of transaction with less valuable terms. There are many reasons this valuation is lower:Willingness to sell. Investors were getting antsy about the IPO. VC’s don’t like telling their LP’s to wait 10 years for a return on their money. So these guys were willing to take a lower valuation to take money off the table.Most investors only sold a portion of their position. Even if SoftBank had offered to buy the whole company at $48B, I doubt investors would have sold much more than they did here. So, by definition, the stock that was exchanged in the tender offer was worth $48B, but the remainder of the shares of the company are likely worth more.Less favorable terms. I believe the stock SoftBank bought has whatever terms were originally associated with the funding round during which they were originally purchased. (And I imagine some of the shares are not even preferred shares at all!) This means in an IPO or exit event, the shares SoftBank bought in this secondary sale could be worth much less than the shares they bought at $68B. This is because preferred shares are generally paid out in reverse chronological order.The second incorrect assumption is about “below-market salaries.” My compensation here is comparable to offers I received from other major tech companies, assuming the the stock grants are worth their 409A valuation price. And Liane, our head of HR, recently gave raises to lots of people to bring everyone up to the industry median or higher.

## What is the answer for the differential equation [math](D^2+4)^2 y = \cos(2x)[/math]?

Since this is a non-homogeneous linear differential equation, we solve this in two parts:a) Homogeneous solution.The characteristic equation is [math](r^2 [/math][math][/math][math]+ 4)^2 = 0[/math], which has [math]r = \pm 2i[/math] (both as double roots). Hence, we obtain[math]y_h = (C_1 x [/math][math][/math][math]+ C_2) \cos(2x) [/math][math][/math][math]+ (C_3 x [/math][math][/math][math]+ C_4) \sin(2x). \tag*{}[/math]b) Particular solution.Due to the right side of the differential equation, we would usually guess that[math]y_p = A \cos(2x) [/math][math][/math][math]+ B \sin(2x)[/math]. However, since this is part of the homogeneous solution, we multiply this by [math]x[/math]. Since [math]Ax \cos(2x) [/math][math][/math][math]+ Bx \sin(2x)[/math] is also part of the homogeneous solution, we multiply by [math]x[/math] one more time:[math]y_p = Ax^2 \cos(2x) [/math][math][/math][math]+ Bx^2 \sin(2x). \tag*{}[/math]Differentiating:[math]Dy_p = (-2Ax^2 [/math][math][/math][math]+ 2Bx) \sin(2x) [/math][math][/math][math]+ (2Ax [/math][math][/math][math]+ 2Bx^2) \cos(2x). \tag*{}[/math][math]D^2 y_p = (-4Bx^2 - 8Ax [/math][math][/math][math]+ 2B) \sin(2x) [/math][math][/math][math]+ (-4Ax^2 [/math][math][/math][math]+ 8Bx [/math][math][/math][math]+ 2A) \cos(2x). \tag*{}[/math][math]D^3 y_p = (8Ax^2 - 24Bx - 12A) \sin(2 x) [/math][math][/math][math]+ (-8Bx^2 - 24Ax [/math][math][/math][math]+ 12 B) \cos(2x). \tag*{}[/math][math]D^4 y_p = (16Bx^2 [/math][math][/math][math]+ 64Ax - [/math][math]48B[/math][math]) \sin(2x) [/math][math][/math][math]+ (16Ax^2 - 64Bx - 48A) \cos(2x). \tag*{}[/math]Since [math](D^2 [/math][math][/math][math]+ 4)^2 y = (D^4 [/math][math][/math][math]+ 8D^2 [/math][math][/math][math]+ 16)y[/math], substituting this into the original differential equation yields[math]\Big[(16Bx^2 [/math][math][/math][math]+ 64Ax - [/math][math]48B[/math][math]) \sin(2x) [/math][math][/math][math]+ (16Ax^2 - 64Bx - 48A) \cos(2x)\Big] [/math][math][/math][math]+ 8 \Big[(-4Bx^2 - 8Ax [/math][math][/math][math]+ 2B) \sin(2x) [/math][math][/math][math]+ (-4Ax^2 [/math][math][/math][math]+ 8Bx [/math][math][/math][math]+ 2A) \cos(2x)\Big] [/math][math][/math][math]+ 16 \Big[Ax^2 \cos(2x) [/math][math][/math][math]+ Bx^2 \sin(2x)\Big] = \cos(2x). \tag*{}[/math]Collecting like terms:[math]-32B \sin(2x) - 32A \cos(2x) = \cos(2x). \tag*{}[/math]Equating like coefficients yields [math]A = -\frac{1}{32}[/math] and [math]B = 0[/math].Hence, a general solution is given by[math]y = \displaystyle (C_1 x [/math][math][/math][math]+ C_2) \cos(2x) [/math][math][/math][math]+ (C_3 x [/math][math][/math][math]+ C_4) \sin(2x) - \frac{1}{32} x^2 \cos(2x). \tag*{}[/math](Double checked the answer on Wolfram Alpha.)

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