Q. 15.0( 2 Votes )
Solve the following equations:
x2dy + y(x + y)dx = 0
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
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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