- #1

- 18

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## Homework Statement

Find orthogonal trajectories for y=cln

^{x}, y=ce

^{x}, y=sin(x)+c

^{x}

## Homework Equations

Simple integration for the most part.

## The Attempt at a Solution

I'm fairly confident on the first two, its the third that's giving me trouble.

First, y=cln(x)

c=[itex]\frac{y}{ln(x)}[/itex]

[itex]\frac{dy}{dx}[/itex]=[itex]\frac{c}{x}[/itex]

[itex]\frac{dy}{dx}[/itex]=[itex]\frac{y}{xln(x)}[/itex]=m

_{1}

[itex]\frac{dy}{dx}[/itex]=[itex]\frac{-xln(x)}{y}[/itex]=m

_{2}

[itex]\int[/itex]-xln(x)dx=[itex]\int[/itex]ydy using IBP

[itex]\frac{-lnxx^2}{2}[/itex]-[itex]\int[/itex][itex]\frac{-x}{2}[/itex]dx

c+[itex]\frac{-lnxx^2}{2}[/itex]+[itex]\frac{x^2}{4}[/itex]=[itex]\frac{y^2}{2}[/itex]

Second, y=ce

^{x}

c=[itex]\frac{y}{e^x}[/itex]

[itex]\frac{dy}{dx}[/itex]=ce

^{x}

[itex]\frac{dy}{dx}[/itex]=[itex]\frac{e^xy}{e^x}[/itex]=y=m

_{1}

[itex]\frac{dy}{dx}[/itex]=[itex]\frac{-1}{y}[/itex]=m

_{2}

[itex]\int[/itex]ydy=[itex]\int[/itex]-1dx

[itex]\frac{y^2}{2}[/itex]+x=c

Third, y=sin(x)+cx

^{2}

c=[itex]\frac{y-sin(x)}{x^2}[/itex]

[itex]\frac{dy}{dx}[/itex]=cos(x)+2xc

[itex]\frac{dy}{dx}[/itex]=cos(x)+[itex]\frac{2(y-sin(x))}{x}[/itex]

I've been unable to make any more progress.