Question

Solve with the orthogonal trajectory:
x y = c

Answer 1

Answer on Question #52873 - Math – Differential Equations

Solve with the orthogonal trajectory:

$$x y = c$$

Solution

$$x y = c \rightarrow \frac{d}{dx}(x y) = \frac{d}{dx}(c) \rightarrow y + x \frac{dy}{dx} = 0 \rightarrow \frac{dy}{dx} = -\frac{y}{x}.$$

Using the fact that perpendicular lines have slopes which are negative reciprocals, solve $\frac{dY}{dx} = -\frac{1}{\frac{dy}{dx}}$.

For the orthogonal trajectories

$$\frac{dY}{dX} = \frac{X}{Y}$$

We usually change back to $x$ and $y$ at this point

$$\frac{dy}{dx} = \frac{x}{y}$$

Separate the variables and integrate

$$y dy - x dx = 0$$ $$\frac{y^2}{2} - \frac{x^2}{2} = C_1$$ $$y^2 - x^2 = C_2$$

or

$$x^2 - y^2 = C$$

where $C$ is an arbitrary real constant.

Trajectories of $x y = c$ and $x^2 - y^2 = C$ are hyperbolas.

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