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:


……(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 the like variables on one side





We know that:


and



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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