# I need to solve an equation of this type ( d

antennense 2022-07-01 Answered
I need to solve an equation of this type
${\left(\frac{dy}{dx}\right)}^{2}+\frac{{y}^{2}}{{b}^{2}}={a}^{2}$
but I don't know how to start Any help would be welcome, Thanks !
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## Answers (1)

Jamarcus Shields
Answered 2022-07-02 Author has 17 answers
${\left(\frac{dy}{dx}\right)}^{2}={a}^{2}-\frac{{y}^{2}}{{b}^{2}}$
This gives us
$\frac{dy}{dx}=±\sqrt{{a}^{2}-\frac{{y}^{2}}{{b}^{2}}}$
$\frac{dy}{\sqrt{{a}^{2}{b}^{2}-{y}^{2}}}=±\frac{dx}{b}$
Setting $y=ab\mathrm{cos}\left(t\right)$, gives us
$\frac{-ab\mathrm{sin}\left(t\right)dt}{ab\mathrm{sin}\left(t\right)}=±\frac{dx}{b}$
This gives us
$dt=\mp \frac{dx}{b}\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}t=\mp \frac{x}{b}+c$
Hence,
$y=ab\mathrm{cos}\left(t\right)=ab\mathrm{cos}\left(\frac{x}{b}+k\right)$

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