Differential Equations

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Presentation transcript:

Differential Equations Chapter 3 Application of First order Linear Differential Equations

Orthogonal trajectories Orthogonal Trajectories Chapter 3 Chapter 3

Orthogonal Trajectories Chapter 3

Orthogonal Trajectories Chapter 3 METHOD Eq.1. Given family of curve STEP 1 Find the differential equation for the given family of curves, by differentiating Eq.1. Eq.2. Orthogonal Trajectories Chapter 3 STEP 2 Find the differential equation of the Orthogonal Trajectory Eq.3. STEP 3 Solution of Eq.3. will be the equation of the family of orthogonal trajectories

Orthogonal Trajectories Chapter 3 Orthogonal Curves (1) Example Solution Orthogonal Trajectories Chapter 3 Two curve intersect perpendicularly if the product of the slopes of the tangents at the intersection point is -1. The differential equation for the orthogonal family of curves.

Orthogonal Trajectories Chapter 3 Orthogonal Curves (2) Example Solution (cont’d) Orthogonal Trajectories Chapter 3 The figure on the right shows these two orthogonal families of curves.

Orthogonal Trajectories Chapter 3 [1/2] Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: STEP 1 Differentiating Eq.1 Orthogonal Trajectories Chapter 3

Orthogonal Trajectories Chapter 3 [2/2] STEP 2 Differential equation of the orthogonal trajectories is Solving the differential equation by method of separation of variables STEP 3 Orthogonal Trajectories Chapter 3 is required equation of orthogonal trajectories .

Orthogonal Trajectories Chapter 3 Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: STEP 1 Differentiating Eq.1 we obtained Orthogonal Trajectories Chapter 3 STEP 2 The differential equation of the Orthogonal Trajectory is STEP 3 Solving this Homogeneous DE, we obtained the equation of family of orthogonal trajectories

Orthogonal Trajectories Chapter 3 Example: Find the equation of the orthogonal trajectories of the family of curves Eq.1. Solution: Differentiating Eq.1 we obtained STEP 1 Orthogonal Trajectories Chapter 3 STEP 2 The differential equation of the Orthogonal Trajectory is STEP 3 Solving this Homogeneous DE, we obtained the equation of family of orthogonal trajectories

Orthogonal Trajectories Chapter 3