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


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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