Differential equation of (x+y)dy/dx=4x+y is

Question

Complete solution 

Answer 1

I can get yo started on this problem.

 

(dy/dx) = (4x + y)/(x+y) = [4 + (y/x)]/[! + (y/x)]

 

Make the substitution y/x = v or y = vx and dy/dx = v + dv/dx

 

v + dv/dx = (4 + v)/(1 +v)

 

dv/dx = [(4 +v)/(1 + v)] - v = (4 - v²)/(1 + v)

 

divide the right side by long division

 

dv/dx = -{v - 1 - [3/(v +1)]}

 

v + C = (-v²/2) + v  + 3 ln(v + 1)

 

3 ln(v +1) - (v²/2) = C  This is the general solution in terms of v

 

You can now convert back to x and y using the relationship v = y/x