# Solve the following equations: Given differential equation can be written as: ……(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 one side we get,  We know that: Integrating on both sides, we get, log(v2 + 1) = -logx + logC ( LogC is an arbitrary constant)

Since y = vx,

we get  ( )

Applying exponential on both sides, we get,  Cross multiplying on both sides we get,

y2 + x2 = Cx

The solution for the given differential equation is y2 + x2 = Cx.

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