Question

У-хр=x-yp please solve the problem
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Answer 1

Answer:

y−xp=x+yp,p=dydx

⇒(x+y)dydx=y−x

⇒dydx=y−xy+x

To solve this equation we put y=vx⇒dydx=v+xdvdx

Then the above equation changes to..

v+xdvdx=vx−xvx+x

⇒xdvdx=v−1−v2−vv+1

⇒−∫(v+1)dvv2+1=∫dxx+C

12∫2vdvv2+1+∫dvv2+1=−∫dxx+C

⇒12ln(v2+1)+arctan(v)=−ln(x)+C

⇒12ln(x2+y2)−ln(x)+arctan(yx)=−ln(x)+C

⇒x2+y2−−−−−−√⋅earctan(yx)=C