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Question

Show that the given differential equation is homogeneous and then solve it.

(x2+xy)dy=(x2+y2)dx.

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Solution

Given, differential equation is dydx=x2+y2x2+xy ...(i)
Here, the numerator and denominator both have polynomial of degree 2. So, the given differential equation is homogeneous.
Let y=vxdydx=v+xdvdx
On putting the values of dydx and y in Eq. (i), we get
v+xdvdx=x2+x2v2x2vx2v+xdvdx=x2(1+v2)x2(1+v)xdvdx=1+v21+vvxdvdx=1+v2vv21+v=1v1+v1+v1vdv=dxx21+v1vdv=dxx
On integarting both sides, we get
21vdv1dv=1xdx2log(1v)v=logxlogCv2log(1v)logx=logCv+2log(1v)+logx=logCPut v=yxyx+2log(1yx)+logx=logCyx+log(xyx)2+logx=logCyx+log(xy)2logx2+logx=logCyx+log(xy)22logx+logx=logCyx+log(xy)2logx=logCyx+log(xy)2xlogC=0log((xy)2x.1C)=yx(xy)2=Cxeyx
This is the required solution of the given differential equation.


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