First order DE prob: find the family of curves that....

Fish314

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The problem that I am trying to solve comes from Tenebaum and Pollard's ORDINARY DIFFERENTIAL EQUATIONS. It's problem 13.8.

The problem as stated is "find the family of curves with the property described: The lenth of a tangent segment from the point of contact to the x intercept is a constant".

Here's how far I've progressed. First, the length of a line segment is equal to SQRT[(x1-x2)^2+(y1-y2)^2]. The two points in question are a point on the curve (from which the tangent would be taken) and the x intercept. Since we are talking about the point of contact on the curve, I'm going to replace x1 and y1 with the just x and y.

y2 is 0, because we are talking about the x intercept and we can use the point slope formula for a line to solve for x2, with the slope of the tangent line being y'. This yields x2=x-y/y'

Substituting in above gives the following differential equation:

SQRT[(y/y')^2+y^2]=k

From this point, I should solve the differential equation, but my problem is that I don't know how to solve an equation when the y' term is raised to a power. Up to this point, the methods for solving D.E.'s that we've gone over are separation of variables, exact equations, homogeneous equations, and integrating factors. I'm inclined to try to separate the variables, (and in fact the equation does separate). Here's how it separates below.

1. Begin by squaring both sides

k^2=(y/y')^2+y^2

2. Subtract y^2 from both sides
k^2-y^2=(y/y')^2

3. Divide by y^2

(k^2-y^2)/y^2=1/y'^2

4. Multiply both sides by dy^2

dy^2*(k^2-y^2)/y^2=dx^2


From here I think I should be able to integrate, but how do I evaluate an integral where the dy or the dx is raised to a power?


Thanks very much for the help.

-Jeremy
 
\(\displaystyle \sqrt{(\frac{y}{y'})^2 + y^2}=k \,\, \Rightarrow \,\,(\frac{y}{y'})^2 + y^2 = k^2 \,\, \Rightarrow \,\, \sqrt{(\frac{k^2-y^2}{y^2})}\,\,dy = \,\,^+_- dx\)
 
Oh! Duh. <-- That would be the sound of me being stupid.

Of course, thanks for the help! God I can't believe I missed that!
 
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