By using the substitution y=vx, find the particular solution of the differential equation 2xy (dy/dx)=y^2-x^2 when y=4 and x=2. Express y in terms of x.?

1 Answer
Mar 15, 2018

y=x10x1

Explanation:

We use the substitution y=vx. This means that

dydx=v+xdvdx

Now, the given equation is

2xy(dydx)=y2x2
or
dydx=12(yxxy)

so that

v+xdvdx=12(v1v)
xdvdx=12(1v+v)
which can be rewritten in the form

2vv2+1dv=dxx

d(v2+1)v2+1=dxx

which readily integrates to

ln(v2+1)=lnx+C

The given initial condition is y=4 at x=2. This means that at x=2, v=42=2. Substituting this in the solution we have obtained so far, we get

ln5=ln2+C

giving the value of the constant of integration as ln10.

The solution is thus

(v2+1)=10x

or

y2x2+1=10x

or

y=x10x1