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=$\int$ydy 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
$\int$ydy=$\int$-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!