Question

How to show that y=3x+(2A/x) is the general solution of the equation x(dy/dx)+y=6x?

Answer 1

Solve first the homogeneous equation:

`xdy/dx + y = 0`

which is separable once we exclude the trivial solution `y=x=0` identically:

`dy/dx =-y/x`

`dy/y =-dx/x`

`int dy/y =-int dx/x`

`ln abs y = - ln abs x + C`

`ln abs y = ln abs (e^C/x)`

`y = c/x`

Now we can use the variable coefficients to look for a specific solution of the complete equation in the form:

`bary = (c(x))/x`

`(dbary)/dx = (xc'(x)-c(x))/x^2`

substituting in the original equation:

`x(dbary)/dx +bary = 6x`

`(xc'(x)-c(x))/x + (c(x))/x = 6x`

`c'(x) =6x`

`c(x) = 3x^2+ c_1`

The general solution is then:

`y= c/x +(3x^2+c_1)/x`

and if we let `A= c+c_1`:

`y= A/x+3x`

Answer 2

see below

Explanation:

`x(dy)/(dx)+y=3x`

by inspection the LHS is the differential of a product rule

`d/(dx)(xy)=6x`

integrating both sides

`xy=3x^2+c`

`=>y=3x+c/x`

`"let " c=2A`

where `A=` a constant

`y=3x+(2A)/x`