Q. 15.0( 2 Votes )

Solve the following equations:

x2dy + y(x + y)dx = 0

Answer :

Let us write the given differential equation in the standard form:


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 the like variables on one side

We know that:


Integrating on both sides we get

( logC is also an arbitrary constant)


( xloga = logax)

Applying exponential on both sides, we get,

Squaring on both sides we get,

Since y = vx

we get

Cross multiplying on both sides we get,

yx2 = c2(y + 2x)

The solution to the given differential equation is yx2 = c2(y + 2x)

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