Solve this DE using homogeneous equations

[BifSlamkovich]

Homework Statement

dy/dx = (6x^(2)+xy+6y^(2))/(x^2)

Homework Equations

v = y/x
y' = v + xv'

The Attempt at a Solution

y' = tan(6ln(abs(x))-C)/x ===> apparently not correct

 

[Curious3141]

Homework Statement

dy/dx = (6x^(2)+xy+6y^(2))/(x^2)

Homework Equations

v = y/x
y' = v + xv'

The Attempt at a Solution

y' = tan(6ln(abs(x))-C)/x ===> apparently not correct

Tough to see where you went wrong if you don't show all your working.

In your final line, I presume you meant y instead of y' (the latter means derivative of y).

Your answer looks almost correct, except that in your last step in going from v to y you may have divided by x instead of multiplying by x. Remember that v = \frac{y}{x}, so y = vx.

Another pointer (not that your post is wrong in this respect, but this makes things neater) is that when you have a ln term plus an arbitrary constant, it's neater to express it as \ln|kx|. This is because \ln|kx| = \ln|x| + \ln|k|, which is equivalent in form to \ln|x| + C.

So your final answer should be y = x\tan(6\ln|kx|), where k is an arbitrary constant.

PS: also, to correct a possible misconception in your thread title, you're not solving the DE "using" homogeneous equations. The DE itself is a homogeneous equation, the definition of which is that multiplying all variables (both x and y in this case) by a constant scaling factor will not change the equation.

 

[HallsofIvy]

You are correct up to the very last step. y= xv, not v/x!