Some Engeneering Applications on ODEs

Some Engineering Applications for first order ODE

Orthogonal Trajectories
Consider a one-parameter family of curves in the xy-plane as:
F(x,y,c)=0
Where c is parameter .We must find another one-parameter family of curves, orthogonal trajectories of the original family given as;
G(x,y,k)=0
Every curve in this new family intersect at right angles every curve in the original family .
We first differentiate with respect to x then eliminate c between this derived equations ,and solve for y to obtain the differential equation as;
dy/dx=f(x,y)
The orthogonal trajectories are the solutions of
dy/dx=(-1)/(fConsider a one-parameter family of curves in the xy-plane as:
F(x,y,c)=0
Where c is parameter .We must find another one-parameter family of curves, orthogonal trajectories of the original family given as;
G(x,y,k)=0
Every curve in this new family intersect at right angles every curve in the original family .
We first differentiate with respect to x then eliminate c between this derived equations ,and solve for y to obtain the differential equation as;
dy/dx=f(x,y)
The orthogonal trajectories are the solutions of
dy/dx=(-1)/(f(x,y))

centered at the origion
Such orthognal systems of curves are particulare importance in physical problems related to electrical potential (flow of electric current and other to curves of constant potential ) , they also occure in hydrodynamics and heat flow problems .
(x,y))
Consider a one-parameter family of curves in the xy-plane as:
F(x,y,c)=0
Where c is parameter .We must find another one-parameter family of curves, orthogonal trajectories of the original family given as;
G(x,y,k)=0
Every curve in this new family intersect at right angles every curve in the original family .
We first differentiate with respect to x then eliminate c between this derived equations ,and solve for y to obtain the differential equation as;
dy/dx=f(x,y)
The orthogonal trajectories are the solutions of
dy/dx=(-1)/(f(x,y))

centered at the origion
Such orthognal systems of curves are particulare importance in physical problems related to electrical potential (flow of electric current and other to curves of constant potential ) , they also occure in hydrodynamics and heat flow problems .




centered at the origion
Such orthognal systems of curves are particulare importance in physical problems related to electrical potential (flow of electric current and other to curves of constant potential ) , they also occure in hydrodynamics and heat flow problems .
Some Engineering Applications for first order ODE
Orthogonal Trajectories