Question

Form the differential equation by eliminating f and ϕ from z = xf(yx)+yϕ(x)

Answer 1

$$z(x, y)=xf(yx)+y\phi(x)$$

$p=z_x=f(yx)+xyf'(yx)+y\phi'(x)$

$q=z_y=x^2f'(yx)+\phi(x)$

$z_{yy}=x^3f''(yx)$

$z_{xy}=2xf'(yx)+x^2yf''(yx)+\phi'(x)$

$z_{xy}=2xf'(yx)+\dfrac{y}{x}z_{yy}+\phi'(x)$

$px+qy=xf(yx)+x^2yf'(yx)+xy\phi'(x)+x^2yf(yx)+y\phi(x)$

$px+qy=z+xy(2xf'(yx)+\phi'(x))$

$px+qy=z+xy(z_{xy}-\dfrac{y}{x}z_{yy})$

$px+qy=z+xyz_{xy}-y^2z_{yy}$