Find a differential equation for w(t)=ϕt(z)=z+tan⁡t1−z(tan⁡t)Usually the question is the other way around, but (D. H. Sattinger and O. L. Weaver, 1986 Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics) pose it like this.Earlier we showed that ϕt+s=ϕt∘ϕs.Since tan is a periodic function, I thing that the differential equation they are looking for might be of order 2.My efforts so far consists of calculating w'(t), and trying to identify w in there, but things get messy.

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Find a differential equation for w(t)=ϕt(z)=z+tan⁡t1−z(tan⁡t)Usually the question is the other way around, but (D. H. Sattinger and O. L. Weaver, 1986 Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics) pose it like this.Earlier we showed that ϕt+s=ϕt∘ϕs.Since tan is a periodic function, I thing that the differential equation they are looking for might be of order 2.My efforts so far consists of calculating w&#039;(t), and trying to identify w in there, but things get messy.

mistemePietsffi · Accepted Answer

Step 1What about something like the following:w(tan⁡(x))=tan⁡(x)+tan⁡(t)1−tan⁡(x)tan⁡(t)&amp;nbsp;=tan⁡(x+t).Step 2Differentiating both sides with respect to x, using that&amp;nbsp;tan⁡&amp;nbsp;′(x)=1+tan2(x), this becomesw′(tan⁡(x))(1+tan2(x))=1+tan2(x+t),i.e.&amp;nbsp;w′(z)(1+z2)=1+(w(z))2.Will this do? In fact, if there are no restrictions on what kind of a differential equation you need, you can probably just multiply the defining equation of w(t) by the denominator and differentiate both sides, for example.

Find a differential equation for \(\displaystyle{w}{\left({t}\right)}=\phi_{{t}}{\left({z}\right)}={\frac{{{z}+{\tan{{t}}}}}{{{1}-{z}{\left({\tan{{t}}}\right)}}}}\) Usually the

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