Question

challenge Qu!

Answer 1

this is from a boat crossing a river question so it's physics :-)


solve:

\(y' = \frac{y}{x} - a \frac{\sqrt{x^2 + y^2}}{x}\)

clue: [my] answer is \(y + \sqrt{x^2 + y^2} = C x^{1-a}\) where C is a constant....


i originally thought it was crying out for polar coordinates but i couldn't get it to work.

Answer 2

Right hand side is homogeneous
so \(y=vx\) substitution works nicely right ?

Answer 3

head in hands, about to start crying.... :-(( took me all morning to do this, please don't do it in 5 minutes!!!

and, yes, that's a start. i used an integrating factor, and y/x sub and then......some hyperbolics!!

Answer 4

There comes a time when one has had enough practice at solving differential equations.

WolfrmAlpha makes the substitution y=vx as suggested by ganeshie8 and then shows the step by step method to give:

y(x) = -x sinh(a log(x) - c)