Question

determine the orthogonal trajectory of families. (illustrate)

1. x^2+y^2+2ay-1=0
2. y=a(e^(-2x))

Answer 1

Answer on Question #79726 - Math - Differential Equations

Determine the orthogonal trajectory of families. (illustrate)

Question

1.

$$x ^ {2} + y ^ {2} + 2 a y - 1 = 0$$

Solution

Differentiate respect to $x$:

$$2 x + 2 y y ^ {\prime} + 2 a y ^ {\prime} = 0$$ $$x + y ^ {\prime} (y + a) = 0$$

Substitute $y^\prime$ to $(-1 / y^{\prime})$

$$x - \frac {(y + a)}{y ^ {\prime}} = 0$$ $$y ^ {\prime} = \frac {(y + a)}{x}$$ $$\frac {d y}{y + a} = \frac {d x}{x}$$ $$y + a = c x$$

Answer:

$$y + a = c x$$

For example: if $a = 0, c = 1$

Question

2.

$$y = a e ^ {- 2 x}$$

Solution

$$y ^ {\prime} = - 2 a e ^ {- 2 x}$$ $$\frac {y}{y ^ {\prime}} = - \frac {1}{2}$$

Substitute $y^\prime$ to $(-1 / y^{\prime})$ ..

$$y y ^ {\prime} = \frac {1}{2}$$ $$2 y d y = d x$$ $$y ^ {2} = x + c$$

Answer:

$$y ^ {2} = x + c$$

For example: if $a = 1, c = 1$

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