Find the orthogonal trajectories of the following curves:
r=a(1+sinθ) ,
Solution:
r′=acosθ ⟹a=cosθr′
then r=cosθr′(1+sinθ) .
For orthogonal trajectories in polar coordinates substitution is used: r′→−r′r2
Then r=−r′r2cosθ1+sinθ ,
dθdr=−rcosθ1+sinθ ,
∫rdr=−∫cosθ1+sinθdθ ,
lnr=−∫cosθ1+sinθdθ ,
∫cosθ1+sinθdθ=∫cos2θ(1+sinθ)cosθdθ=∫1−sin2θ(1+sinθ)d(sinx)=∫1−sinxd(sinx)
−∫1−sinxd(1−sinx)¬=−ln(1−sinx)+lnC .
Answer:
r=С(1−sinθ)
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