Question

Which of the following is a homogeneous differential equation?
(a) (4x +6y+5)dy-(3y+2x+4)dx=0
(b) $xydx-(x^3+y^3)dy=0$
(c) $(x^3+2y^2)dx+2xydy=0$
(d) $y^{2}dx+(x^2-xy-y^2)dy=0$

Answer 1

(a) Given equation (4x +6y+5)dy-(3y+2x+4)dx=0 can be rewritten as
$\frac{dy}{dx}=\frac{3y+2x+4}{4x+6y+5}\Rightarrow \frac{dy}{dx}=\frac{x(3.\frac{y}{x}+2+\frac{4}{x})}{x(4+6\frac{y}{x}+\frac{5}{x})}$
This equation is not homogeneous.
(b) Given equation is $xydx-(x^3+y^3)dy=0$
On separating the variables, we get
$(x^3+y^3)dy=xydx\Rightarrow \frac{dy}{dx}=\frac{xy}{x^3+y^3}\Rightarrow \frac{dy}{dx}=\frac{\frac{y}{x^2}}{1+{\left(\frac{y}{x}\right)}^3}$
This is not homogeneous equation.
(c) Given equation is $(x^3+2y^2)dx+2xydy=0$
$2xydy=-(x^3+2y^2)dx$
On separating the variables, we get
$\frac{dy}{dx}=-\frac{x^3+2y^2}{2xy}\Rightarrow \frac{dy}{dx}=\frac{1+\frac{2y^2}{x^3}}{\frac{2y}{x^2}}$
This equation is not homogeneous equation.
(d) Given equation $y^{2}dx+(x^2-xy-y^2)dy=0$
Separating the variable
$(x^2-xy-y^2)dy=-y^{2}dx\Rightarrow \frac{dy}{dx}=-\frac{y^2}{x^2-xy-y^2}\Rightarrow \frac{dy}{dx}=-\frac{{\left(\frac{y}{x}\right)}^2}{(1-\frac{y}{x}-{\left(\frac{y}{x}\right)}^2)}$
This equation is homogeneous equation. So, (d) is the correct option.