alleryTo solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:
A$y - x = v$
B$x + y = v$
C$xy = v$
D$y = vx$
Answer & Solution
Correct answer: D. $y = vx$
The right side is a homogeneous function of degree zero in $x$ and $y$, so the equation has the form $dy/dx = F(y/x)$.
Substituting $y = vx$ converts it into a variables-separable equation in $v$ and $x$ via $\dfrac{dy}{dx} = v + x\dfrac{dv}{dx}$.
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