Question

Find the linear substitution that reduces the inhomogeneous equation to the homogeneous one: (8y + 10x)dx +(7x +5y)dy = 0.

Answer 1

$$(8y + 10x)dx +(7x +5y)dy = 0$$

$$(7x +5y)y' =-(8y + 10x)$$

$$y'=-\dfrac{8y+10x}{7x+5y}$$

Use the linear substitution  $y=vx, v$ is a function of $x$

$$y'=v+v'x$$

$$v+v'x=-\dfrac{8vx+10x}{7x+5vx}$$

The homogeneous

$$v'x=-\dfrac{13v+17}{7+5v}$$

$$\dfrac{5v+7}{13v+17}=-\dfrac{dx}{x}$$