Alternative form of y=11−(xa)2?I solved the differential equation y′(x)=2a2xy2(x) (with y(0)=1 and x∈(−|a|,|a|)) and came toy(x)=11−(xa)2.This looks like some trigonometric thing to me. Do you know a better form for y?

You might think oftanh′(x)=11−x2.So you gety(x)=11−(xa)2=tanh′(xa)

Expert answers to "Alternative form of y=11−(xa)2?I solved the differential equation..."

fellesturduj

2022-03-17

Alternative form of

y = \frac{1}{1 - {(\frac{x}{a})}^{2}}

?
I solved the differential equation

y^{'} (x) = \frac{2}{a^{2}} x y^{2} (x)

(with y(0)=1 and

x \in (- | a |, | a |)

) and came to

y (x) = \frac{1}{1 - {(\frac{x}{a})}^{2}} .

This looks like some trigonometric thing to me. Do you know a better form for y?

Deven Paul

Beginner2022-03-18Added 7 answers

You might think of

tanh^{'} (x) = \frac{1}{1 - x^{2}} .

So you get

y (x) = \frac{1}{1 - {(\frac{x}{a})}^{2}} = tanh^{'} (\frac{x}{a})

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