r=a(1+cosθ) Differentiate
dr=−asinθdθ
a=1+cosθr
dr=−1+cosθrsinθdθ
−r1dθdr=1+cosθsinθThis is the differential equation of the direction field D1 for the given family F1. To find the differential equation of the direction field orthological to D1, we replace dθdr by −r2drdθ.
The differential equation for the orthogonal trajectories becomes
rdrdθ=1+cosθsinθ=2cos2(2θ)2sin(2θ)cos(2θ)=tan(2θ) which is a case of variable-separable, and on integration gives
lnr=2ln(sin(2θ))+ln(2c)This is the equation of the orthogonal family F2.
Since r=c(1–cosθ) represents the same curve as r=c(1+cosθ), the member of F2 with label c is the same as the member of F1 with label a=–c. Thus the given family is selforthogonal.
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