Answer to Question #91477 in Differential Equations for Sajid

Question

Q.Which of the following equations is homogenous
i) (x2+3xy+y2)dx+x2dy=0
ii) dy/dx=(x^2 (x-y)(x+y))/(x^4+y^4 )
iii) dy/dx=((x-y+1)(x+y+1))/(x^2+y^2 )
a. i) only
b. i) and ii)
c. ii) and iii)
d. i) and iii)

Answer 1

Differential equation is homogenous when

$$\frac{dy}{dx}=f(x,y)=f(tx,ty)$$

i)$\frac{dy}{dx}=-1-3\frac{y}{x}-\frac{y^2}{x^2}$ - homogenous;

ii)$\frac{dy}{dx}=\frac{x^2(x-y)(x+y)}{x^4+y^4}$ - homogenous;

iii)$\frac{dy}{dx}=\frac{(x-y+1)(x+y+1)}{x^2+y^2}$ - heterogeneous.

Answer is b) i) and ii).