Can someone solve this?

Question

392 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-05-07T00:46:00

#x \ dy - y \ dx = sqrt(x^2 + y^2) dx#

#implies \ y' - y/x = sqrt(1 + (y/x)^2) #

Let: #z = y/x qquad qquad z' x + z = y'#

#implies \ z' x + z - z = sqrt(1 + z^2) #

#implies \ (z')/sqrt(1 + z^2) = 1/x #

#implies \int \ (dz)/sqrt(1 + z^2) = int \ (dx)/x #

You can prove this for yourself:

#int 1/sqrt(1 + z^2) \ dz = sinh^(-1)(z) + C#

#implies \ sinh^(-1) z = ln|x| + C #

# z = sinh (ln|x| + C ) #

#implies y = x ( sinh (ln|x| + C ) )#