- #1
jimmypoopins
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Hey all, i think I'm doing most of this right, but I'm missing a coefficient somewhere when integrating or something...
Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation.
y'=(3y^2-x^2)/(2xy)
v=y/x
y=xv(x) => y'=v+xv'
v'=dv/dx
y'=(3y^2-x^2)/(2xy)
divide top and bottom of right hand side by x^2 to get v's and replace y' by v+xv'
v+xv'=(3v^2-1)/(2v)
subtract v from both sides
xv'=(3v^2-1)/(2v)-v
put the lonely v on a common denominator
xv'=(3v^2-1-2v^2)/(2v)=(v^2-1)/(2v)
separate v's and x's
(2v)dv/(v^2-1)=dx/x
integrate
ln|v^2-1|=ln|x|+c
substitute v=y/x
ln|(y/x)^2-1|=ln|x|+c
simplify
ln|y^2-x^2|=ln|x|+c
the back of the book says the answer is |y^2-x^2|=c|x|^3.
what am i doing wrong? I'm missing a 3 somewhere. I'm kinda rusty with lograthimic algebra, so all help is appreciated. I've gotten a few of these problems wrong by missing a constant or exponent on the right hand of the side of the equation after integrating.
Homework Statement
Substitute v=y/x into the following differential equation to show that it is homogeneous, and then solve the differential equation.
y'=(3y^2-x^2)/(2xy)
Homework Equations
v=y/x
y=xv(x) => y'=v+xv'
v'=dv/dx
The Attempt at a Solution
y'=(3y^2-x^2)/(2xy)
divide top and bottom of right hand side by x^2 to get v's and replace y' by v+xv'
v+xv'=(3v^2-1)/(2v)
subtract v from both sides
xv'=(3v^2-1)/(2v)-v
put the lonely v on a common denominator
xv'=(3v^2-1-2v^2)/(2v)=(v^2-1)/(2v)
separate v's and x's
(2v)dv/(v^2-1)=dx/x
integrate
ln|v^2-1|=ln|x|+c
substitute v=y/x
ln|(y/x)^2-1|=ln|x|+c
simplify
ln|y^2-x^2|=ln|x|+c
the back of the book says the answer is |y^2-x^2|=c|x|^3.
what am i doing wrong? I'm missing a 3 somewhere. I'm kinda rusty with lograthimic algebra, so all help is appreciated. I've gotten a few of these problems wrong by missing a constant or exponent on the right hand of the side of the equation after integrating.