3.2.1 & 3.3 Orthogonal Trajectories and Applications of Seperable Differential Equations

3.2.1 Orthogonal Trajectories

• Finding parallel and perpendicular vector in terms of another vector

Example 14. Find the family of orthogonal trajectories corresponding to the family of curves $y=kx^4$ and sketch several members of each family

Step 1) Differentiate the equation for the family of curves to get the differential equation that gives the slope of its tangent lines

• $y=kx^4$

• $\frac {dy}{dx} = k(4x³)$

Step 2) Substitute $k$

• $y = kx^4 \Rightarrow k = \frac y{x^4}$

• $\Rightarrow \frac {dy}{dx} = \frac {y4x³}{x^4}$

• $\Rightarrow \frac {dy}{dx} = \frac {4y}x$

Step 3) Find the negative reciprocal to find the differential equation for the corresponding family of orthogonal trajectories

• $\frac {dy}{dx} = \frac {4y}{x} \Rightarrow \frac {dy}{dx} = - \frac x{4y}$

• $\Rightarrow 4ydy=-xdx$

• $\int 4ydy = \int -xdx$

• $4\frac {y²}{2} = -\frac {x²}2 + C_1$

• $x² + 4y² = C$

This result is a family of ellipses (centers at the origin, major axes on the x-axis.)