91Ó°ÊÓ

A differential equation is separable if it can be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \), where \( g(x) \) is a function of \( x \) only, and \( h(y) \) is a function of \( y \) only.

Begin checking each differential equation to see if it can be written in the form \( \frac{dy}{dx} = g(x)h(y) \).(a) \( y' = y \) is separable: \( y' = 1 \cdot y \), so \( g(x)=1, h(y)=y \).(b) \( y' = x + y \) is not separable because it can't be split into \( g(x) \) and \( h(y) \).(c) \( y' = xy \) is separable: \( y' = x \cdot y \), so \( g(x)=x, h(y)=y \).(d) \( y' = \sin(x+y) \) is not separable due to the mixed term inside the sine function.(e) \( y' - xy = 0 \) can be rewritten as \( y' = xy \): is separable, \( g(x)=x, h(y)=y \).(f) \( y' = \frac{y}{x} \) is separable: \( y' = \frac{1}{x} \cdot y \), so \( g(x)= \frac{1}{x}, h(y)=y \).(g) \( y' = \ln(xy) \) is not separable due to the logarithmic mixed term.(h) \( y' = \sin(x)\cos(y) \) is separable: \( y' = (\sin(x)) \cdot (\cos(y)) \), so \( g(x)=\sin(x), h(y)=\cos(y) \).(i) \( y' = \sin(x)\cos(xy) \) is not separable because of the mixed term in the cosine function.(j) \( y' = \frac{x}{y} \) is separable: \( y' = x \cdot \frac{1}{y} \), so \( g(x)=x, h(y)=\frac{1}{y} \).(k) \( y' = 2x \) is separable: \( y' = 2x \cdot 1 \), so \( g(x)=2x, h(y)=1 \).(l) \( y' = \frac{x+y}{x+2y} \) is not separable because the function cannot be expressed as \( g(x)h(y) \).