Orthogonal Trajectory. Calc IV.

[Gummy Bear]

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=\frac{y}{ln(x)}
\frac{dy}{dx}=\frac{c}{x}
\frac{dy}{dx}=\frac{y}{xln(x)}=m₁
\frac{dy}{dx}=\frac{-xln(x)}{y}=m₂
\int-xln(x)dx=\intydy using IBP
\frac{-lnxx^2}{2}-\int\frac{-x}{2}dx
c+\frac{-lnxx^2}{2}+\frac{x^2}{4}=\frac{y^2}{2}

Second, y=ce^x
c=\frac{y}{e^x}
\frac{dy}{dx}=ce^x
\frac{dy}{dx}=\frac{e^xy}{e^x}=y=m₁
\frac{dy}{dx}=\frac{-1}{y}=m₂
\intydy=\int-1dx
\frac{y^2}{2}+x=c

Third, y=sin(x)+cx²
c=\frac{y-sin(x)}{x^2}
\frac{dy}{dx}=cos(x)+2xc
\frac{dy}{dx}=cos(x)+\frac{2(y-sin(x))}{x}

I've been unable to make any more progress.

 

[Gummy Bear]

Can nobody give me an idea of where I should go from here? I don't know if I'm having a simple algebra problem or what.

 

[Gummy Bear]

Why won't anyone help me?! I need to know how to solve this problem before class!