Q. 95.0( 1 Vote )

# Solve the following equations:

Given Differential equation is:

……(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 on the same side we get,

We know that

Integrating on both sides we get,

( logC is an arbitrary constant)

log(1-2v2) = -4logx + 4logC

log(1–2v2) = -logx4 + logC4

( xloga = logax)

( )

Applying exponential on both sides we get,

Since y = vx, we get,

Cross multiplying on both sides we get,

x2(x2–2y2) = c4

x4–2x2y2 = c4

The solution for the given differential equation is x4–2x2y2 = C4.

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