Question

Q. Find an equation of orthogonal trajectory of the curve of each of the following:
(i) x=cy2
(ii) x2+y2=cx
(iii) y=ecx
(iv)xy=c
(v)y2=x2+cx

Answer 1

Answer on Question #64582 – Math – Differential Equations

Question

Find an equation of orthogonal trajectory of the curve of each of the following:

i)

$$x = C y ^ {2}$$

Solution

Differentiating $x = Cy^2$ with respect to $x$

$$1 = 2 C y \frac {d y}{d x},$$

where $\frac{dy}{dx} = y'$ .

Substituting $y^\prime \rightarrow (-1 / y^\prime)$

$$1 = 2 C y \cdot \frac {1}{- \frac {d y}{d x}}$$ $$\frac {d y}{d x} = - 2 C y.$$

Integrating both sides

$$- 2 C \int d x = \int \frac {d y}{y},$$ $$- 2 C x + k _ {1} = \ln | y |,$$ $$- 2 C x + l n k _ {2} = l n | y |,$$ $$- 2 C x = l n \frac {| y |}{k _ {2}},$$ $$e ^ {- 2 C x} = \frac {| y |}{k _ {2}},$$ $$y = k e ^ {- 2 C x}.$$

Answer: $y = k e^{-2Cx}$ .

Question

Find an equation of orthogonal trajectory of the curve of each of the following: ii)

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

Solution

Differentiating $x^{2} + y^{2} = Cx$ with respect to $x$ :

$$2 x + 2 y \frac {d y}{d x} = C.$$

Substituting $y^\prime \rightarrow (-1 / y^\prime)$

$$2 x - 2 y \frac {d x}{d y} = C,$$ $$2 x - C = 2 y \frac {d x}{d y},$$ $$d y (2 x - C) = 2 y d x.$$

Integrating both sides

$$\int \frac {d y}{2 y} = \int \frac {d x}{2 x - C},$$ $$\frac {1}{2} \ln | 2 y | = \frac {1}{2} \ln | (2 x - C) |,$$ $$y = x + k \text{ or } y = - x + k.$$

Answer: $y = x + k$ ; $y = -x + k$ .

Question

Find an equation of orthogonal trajectory of the curve of each of the following:

iii)

$$y = e ^ {C x}$$

Solution

Differentiating $y = e^{Cx}$ with respect to $x$

$$\frac {d y}{d x} = C e ^ {C x}$$

Substituting $y^\prime \rightarrow (-1 / y^\prime)$

$$\frac {- 1}{\frac {d y}{d x}} = C e ^ {C x},$$ $$- \frac {d x}{d y} = C e ^ {C x},$$ $$- e ^ {- C x} d x = C d y.$$

Integrating both sides

$$- \int e ^ {- C x} d x = C \int d y,$$ $$\frac {1}{C} e ^ {- C x} + k _ {1} = C y,$$ $$e ^ {- C x} + k _ {2} = C ^ {2} y,$$ $$y = \frac {e ^ {- C x}}{C ^ {2}} + k.$$

Answer: $y = \frac{e^{-Cx}}{C^2} + k$ .

Question

Find an equation of orthogonal trajectory of the curve of each of the following:

iv)

$$x y = C$$

Solution

Differentiating $xy = C$ with respect to $x$

$$y + x \frac{dy}{dx} = 0$$

Substitute $y' \rightarrow (-1/y')$

$$y + x \cdot \frac{-1}{\frac{dy}{dx}} = 0,$$ $$y - x \frac{dx}{dy} = 0,$$ $$x d x = y d y.$$

Integrating both sides

$$\int y d y = \int x d x,$$ $$\frac{y^2}{2} = \frac{x^2}{2} + C_1,$$ $$y^2 - x^2 = k.$$

Answer: $y^2 - x^2 = k$.

Question

Find an equation of orthogonal trajectory of the curve of each of the following: v)

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

Solution

Differentiating $y^{2} = x^{2} + Cx$ with respect to $x$

$$2 y \frac {d y}{d x} = 2 x + C.$$

Substitute $y^\prime \rightarrow (-1 / y^\prime)$

$$2 y \frac {- 1}{\frac {d y}{d x}} = 2 x + C,$$ $$- 2 y \frac {d x}{d y} = 2 x + C,$$ $$- 2 \frac {d x}{2 x + C} = \frac {d y}{y}.$$

Integrating both sides

$$- 2 \int \frac {d x}{2 x + C} = \int \frac {d y}{y},$$ $$- \ln | 2 x + C | + \ln k _ {1} = \ln | y |,$$ $$\ln \frac {k _ {1}}{| 2 x + C |} = \ln | y |,$$ $$y = \frac {k}{2 x + C}.$$

Answer: $y = \frac{k}{2x + C}$ .

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