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Question

Solve the differential equation
dydx=3y7x+73x7y3

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Solution

dydx=7x+3y+73x7y3

The point of intersection of 7x+3y+7=0 and 3x7y3=0 is (1,0)
Let X=x1 and Y=y

we get dYdX=7X+3Y3X7Y

Let Y=kX , we get k+XdkdX=3k737k
XdkdX=7(k21)37k

7dXX=(3k217kk21)dk

By applying integral we get 7lnX=12ln(k1k+1)72ln(k21)=12ln(YXY+X)72ln(Y2X2X2)

Now put Y=y and X=x1 , we get 7ln(x1)=12ln(yx+1y+x1)72ln(y2(x1)2(x1)2)

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