solve the differential equation dy/dx = (r√(x^2+y^2) + by) / bx, for y. The hint is to let v = y/x

Question

By letting v = y/x, transform the differential equation to a
differential equation in v and x. Find the implicit solution of the resulting separable
equation. Then use the relationship y = xv to obtain

y+√(x^2+y^2) = x^(k+1)

Answer 1

The differentiable equation is not separable, so use the variable substitution; v = y/x

 

Switch it around to: y = vx ; substitute vx in for y

Derivative of y = vx gets subbed in for dy/dx

dy/dx = (dv/dx)x + v

 

dy/dx = (r√(x²+y²) + by) / bx

 

(dv/dx)x + v = r(x²+(vx)²)^1/2 /bx +bvx/bx

 

(dv/dx)x + v = r(x²+v²x²)^1/2/bx + v

 

(dv/dx)x = rx(v²+1)^1/2 /bx

 

(dv/dx)x = (r/b)(v²+1)^1/2

 

It is now separable

 

(v²+1)^-1/2 dv = (r/b)(1/x) dx

 

Integrate

 

∫ (v²+1)^-1/2 dv = r/b ∫ 1/x dx

 

ln|v+(v²+1)^1/2| + c = (r/b)ln|x| + c

 

consolidate constants ; sub in y/x for v

 

ln|y/x + (y² + x²)^1/2 /x| = (r/b)ln|x| + c

 

take it to the power of e

 

y/x + (y²+ x²)^1/2 /x = e^cx ^r/b

 

e^c is just another constant; so is r/b which looks to be k in the solution.

multiply both sides by x

 

y + (y²+ x²)^1/2 = C*x*x^k

 

y + (y² + x²)^1/2 = Cx^(k+1)

 

I think there should be a constant multiplied by x^(k+1)

but otherwise appears to be correct. I hope this helps.

 

 

 

 

Answer 2

But the way you have it written
is not what the solution is supposed to
be. Please edit and repost the correct

problem.

 

As written, you will get imaginary solutions.

And it WORKS!!!!

dy/dx = sqrt( x^2 + y^2) + dy)/dx

dy = sqrt( x^2 + y^2) + dy

0 = sqrt( x^2 + y^2)

0 = sqrt ( 1 + y^2/x^2)

0 = sqrt ( 1 + v^2) where V = y/x

0 = 1 + v^2

v^2 = -1

v = +or- i


y/x = i

y = i*x

dy/dx = i

dy = i*dx

Check: sqrt ( x^2 + y^2) + dy =

sqrt ( x^2 + (i*x)^2) + i*dx

sqrt( x^2 - x^2) + i*dx

i*dx = dy