Q. 54.5( 2 Votes )

Solve the following equations:

(x2 – y2)dx – 2xydy = 0

Answer :

Given differential equation is:


(x2 – y2)dx – 2xydy = 0


(x2 – y2)dx = 2xydy


……(1)


Homogeneous equation: A equation is said to be homogeneous if f(zx,zy) = znf(x,y) (where n is the order of the homogeneous equation).


Let us assume






f(zx,zy) = z0f(x,y)


So, given differential equation is a homogeneous differential equation.


We need a substitution to solve this type of linear equation and the substitution is y = vx.


Let us substitute this in (1)



We know that:









Bringing Like variables on same sides we get,





We know that:



Integrating on both sides, we get,




( logC is an arbitrary constant)


Multiplying with -3 on both sides we get,


log|1-3v2| = -3logx + 3logC



( )



( alogx = logxa)



Applying exponential on both sides we get,



Since y = vx, we get,







Cross multiplying on both sides we get,


x(x2 – 3y2) = c3


x3 – 3xy2 = K (say any arbitrary constant)


The solution for the differential equation is x3 – 3xy2 = K


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