Q. 95.0( 1 Vote )

Solve the following equations:


Answer :

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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