Answer to Question #204131 in Differential Equations for Ryukjak

Question

Instructions:

Good day! Equations are given below. Please determine which of the following equation is not a homogeneous equation. Thank you!

Choose the letter.

a. (y² - xy) dx + x²dy = 0

b. y' = 2xy e ^x/y / x²+y² sin(x/y)

Letter b is in fraction form.

c. 3x²ydx + (y+x³) dy = 0

d. 2x²y+3y³ (x³ + 2xy²) y' = 0

Answer 1

A function f(x,y) is called homogeneous of degree n if f(tx,ty)=tⁿf(x,y), where t is independent of x and y and t belongs to real numbers.

Option (c) is not homogeneous.

$3x^2ydx+(y+x^2)dy=0\\ y'=\frac{-3x^2y}{x^3+y}\\ put x=tX and y=tY, we \space get\\ Y'=\frac{t^3(-3X^2Y)}{t(t^2X^3+Y)}\neq t^nf(x,y)\\ \text{And other options are homogeneous.}$

bsns