Question

Let $f(x,y,c_1)=0$ and $f(x,y,c_2)=0$ define two integral curves of a homogeneous first order differential equation. If $P_1$ and $P_2$ are respectively the points of intersection of these curves with an arbitrary line, $y=mx$ then prove that the slopes of these two curves at $P_1$ and $P_2$ are equal.

Answer 1

Let $f(x,y,c_1)=0$ and $f(x,y,c_2)=0$ define two integral curves of a homogeneous first order differential equation. If $P_1$ and $P_2$ are respectively the points of intersection of these curves with an arbitrary line, y = mx then prove that the slopes of these two curves at $P_1$ and $P_2$ are equal?

$Letf(x,y,c_1)=0$ is $x+y+c_1=0$

$f(x,y,c_2)=0$ is $x+y+c_2=0$

i.e the curve is different by constant only let take example of graph

So, slope of the curve at $P_1$ is differentiate $x+y+$$c_1$=0

$1+\frac{dy}{dx}=0$

${\left(\frac{dy}{dx}\right)}_{P_1}=-1$

and slope of the curve at $P_2$ is differentiate $x+y+$$c_2$=0

$1+\frac{dy}{dx}=0$

${\left(\frac{dy}{dx}\right)}_{P_2}=-1$

${\left(\frac{dy}{dx}\right)}_{P_2}$$={\left(\frac{dy}{dx}\right)}_{P_1}$

Hence Proved.