Understanding First Order ODEs and Intersection of Curves

[ka_reem13]

Homework Statement

(a) Solve the differential equation:

[x * (dy/dx)^2] - [2y*(dy/dx)] - x = 0

How many integral curves pass through each point of the (x,y) plane (except x = 0)?
why is the solution at each point not unique

(b) The differential equation:
[(dy/dx)^2] + [f(x,y)*(dy/dx)] - 1 = 0
represents a set of curves such that two curves pass through any given point. Show that these curves intersect at right angles at the point. at f = -2y/x verify this property for the point (3,4)

Relevant Equations

differential equations

I'm aware that I can introduce the perimeter p = dy/dx
then I can rearrange my equation to make y the subject, then I can show that dp/dx = p/x. However, this only gives me a bunch of quadratic curves for my solution. However given part b I see that two curves are meant to intersect each point and I don't know where I'll get the second set of curves (solutions) from.

for part b honestly I don't even know where to start

 

[pasmith]

(a) The ODE is a quadratic in \dfrac{dy}{dx}. How many real roots does it have?

(b) If two lines y = m_1x + c_1 and y = m_2x + c_2 intersect, then the angle between them at the intersection is given by \cos \theta = \frac{(1,m_1)\cdot(1,m_2)}{\|(1,m_1)\|\|(1_,m_2)\|} = \frac{1 + m_1m_2}{\sqrt{1 + m_1^2}\sqrt{1 + m_2^2}}. What is \cos \theta if the lines intersect at right angles? To apply this to two curves, one looks at the tangent lines at the point of intersection. What are the gradients of these tangent lines if the curves are the integral curves of this ODE?

 

[haruspex]

introduce the perimeter

parameter

 

[haruspex]

I can show that dp/dx = p/x

You can? I don’t see how. Please post your working.

 

[Mark44]

I can show that dp/dx = p/x

You can? I don’t see how.

Easy, just cancel the d's.
$$\frac{dp}{dx} = \frac{\cancel dp}{\cancel dx}$$