Trajectories

 Trajectories

 

If any curve cuts, every member curve of any family of curves, according to a rule then that curve is called trajectory of family of curves.

If curve cuts all remaining members of its own family at right angel then it is called self-orthogonal.

Finding orthogonal trajectory-

 Let equation family of curves $f(x,y,\frac{dy}{dx})=0$

·        Differentiate this equation with respect to $\mathcal x$

·        Put $-\frac{dx}{dy}$ instead of $\frac{dy}{dx}$

 

Polar form – $f(r,\theta,\frac{dr}{d\theta})=0$

•          Differentiate with respect to $\theta$
•         Put $-\frac{1}{r}\frac{dr}{d\theta}$  instead $r\frac{d\theta}{dr}$ of & $-r^2\frac{d\theta}{dr}$  instead of  $\frac{dr}{d\theta}$

Also read Linear ordinary differential equations