Question

Which of the following is a homogeneous differential equation?

• A. (4x + 6y + 5) dy - (3y + 2x + 4) dx = 0
• B. $xy \, dx - (x^3 + y^3) dy = 0$
• C. $(x^3 + 2y^2) dx + 2xy dy = 0$
• D. $y^2 dx + (x^2 - xy - y^2) dy$

Answer 1

Answer: (D)

$y^2 dx + (x^2 - xy - y^2) dy$

Answer 2

Consider the differential equation $y^2 dx + (x^2 - xy - y^2) dy = 0$ $\therefore \frac{dy}{dx} = - \frac{y^{2}}{x^{2} -xy -y^{2}} = \frac{y^{2}}{x^{2} +xy -x^{2}}$ $= f\left(x,y\right) \Rightarrow f\left(x,y\right) = \frac{y^{2}}{x^{2} + xy - x^{2}}$ Replacing x by $\lambda x$ and $y$ by $\lambda y \, f(\lambda x , \lambda y)$ $= \frac{\left(\lambda y\right)^{2}}{\left(\lambda x\right)^{2} +\left(\lambda x\right)\left(\lambda y\right) - \left(\lambda x\right)^{2}}$ $= \lambda^{0} \frac{\lambda^{2}y^{2}}{\lambda^{2}y^{2} + \lambda^{2} xy - \lambda^{2}x^{2}}$ $= \lambda^{0} \left(\frac{y^{2} }{y^{2} + xy-x^{2}}\right)$ $= \lambda^{0} f\left(x,y\right)$ $\therefore$ f(x, y) is the homogeneous function of degree zero.