Question

# If the general solution for differential equation dvdx=x+2y−32x+y−3 is |x+y−2|=c|(x−y)n| c > 0 then n = ___

Solution

## x+2y−32x+y−3 would have been a homogeneous function of degree zero if the 3  in numerator and denominator were not there. Our aim is to make constant terms zero. Then it will be a homogeneous function of degree zero and hence the equation will be homogeneous differential equation. Put x = X + h                                       ….(1) y = Y + k                                                ….(2) So,dydx=dYdxSo,dYdx=X+2Y+(h+2k−3)2X+Y(2h+k−3) Now for constant terms to be zero h + 2k - 3 = 0 and 2h + k - 3 should be equal to 0 Solving these 2 equations simultaneously we get h = k =1. So for h = k = 1. dYdX=X+2Y2X+Y which is a homogeneous differential equation. Put Y=vX         so dYdX=v+XdvdX⇒v+Xdvdx=1+2v2+v⇒(2+v1−v2)dv=dxx  which can be solved by variable separable method. ⇒[32(1−v)+12(1+v)]dv=dxx⇒−32log|1−v|+12log|1+v|=log|X|+12logc.⇒log∣∣∣1+v(1−v)3∣∣∣=log x2+log c  ⇒∣∣∣1+v(2(1−v)3)∣∣∣=cX2c>0⇒|Y+X|=c|(Y−X)3|⇒|x+y−2|=c|(x−y)3.|, c > 0 (Putting values of x and y from (1) and (2).  So n=3.

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