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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=vx⇒dydx=v+xdvdx On putting the values of dydx and y in Eq. (i), we get v+xdvdx=x2+x2v2x2vx2⇒v+xdvdx=x2(1+v2)x2(1+v)⇒xdvdx=1+v21+v−v⇒xdvdx=1+v2−v−v21+v=1−v1+v⇒1+v1−vdv=dxx⇒2−1+v1−vdv=dxx On integarting both sides, we get ∫21−vdv−∫1dv=∫1xdx⇒−2log(1−v)−v=logx−logC⇒−v−2log(1−v)−logx=−logC⇒v+2log(1−v)+logx=logCPut v=yx⇒yx+2log(1−yx)+logx=logC⇒yx+log(x−yx)2+logx=logCyx+log(x−y)2−logx2+logx=logCyx+log(x−y)2−2logx+logx=logC⇒yx+log(x−y)2−logx=logC⇒yx+log(x−y)2x−logC=0⇒log((x−y)2x.1C)=−yx⇒(x−y)2=Cxe−yx This is the required solution of the given differential equation.

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