1. Differential Equations
Chapter 3
Application of
First order Linear
Differential Equations

2. Orthogonal trajectories
Orthogonal Trajectories Chapter 3
Chapter 3

3. Orthogonal Trajectories Chapter 3

4. 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

5. 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.

6. 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.

7. 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

8. 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 .

9. 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

10. 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

11. Orthogonal Trajectories Chapter 3