Test if the following ODEs are exact, and if so, solve: 7(i) y′ = yx

The given ordinary differential equation (ODE) is y′ = yx.

First, we need to rewrite this equation in the standard form of an exact differential equation, which is M(x, y)dx + N(x, y)dy = 0.

The given equation can be rewritten as -yx dy + y dx = 0, where M(x, y) = -yx and N(x, y) = y.

An exact differential equation satisfies the condition ∂M/∂y = ∂N/∂x.

Let's check if this condition is satisfied:

∂M/∂y = -x and ∂N/∂x = 0.

Since ∂M/∂y ≠ ∂N/∂x, the given differential equation is not exact.

Therefore, we cannot solve it using the method for exact differential equations.