Question

Obtain the complete integral xp^2- ypq + y^3q - y^2z= 0

Answer 1

Using charpit's auxilliary equation,

$\frac{dp}{∂f/∂x + p(∂f/∂z)}=\frac{dq}{∂f/∂y + q(∂f/∂z)}=\frac{dz}{−p(∂f/∂p)− q(∂f/∂q)}=\frac{dx}{−∂f/∂p}=\frac{dy}{−∂f/∂q}\\ \frac{∂f}{∂x}=p^2\\ \frac{∂f}{∂z}=-y^{2}\\ \frac{∂f}{∂y}=-pq+3qy^2-2zy\\ \frac{∂f}{∂p}=2xp-yq\\ \frac{∂f}{∂q}=-yp+y^3\\ \text{so by substitution}\\ \frac{dp}{p^2-py^2}=\frac{dq}{2qy^2-pq-2zy}=\frac{dz}{-2xp^2+2ypq-qy^3}=\frac{dx}{yq-2xp}=\frac{dy}{yp-y^3}\\ \frac{dp}{p}-\frac{dy}{y}=0\\ log p-logy=log a\\ log p=log a+logy\\ p=ay\\ \text{substitute the value of p in the equation and we have q to be}\\ x({ay})^2-y(ay)q+q({ay})^3-zy^2=0\\ q=\frac{z-xa^2}{ay-a}\\ \text{So we use}\\ dz=pdx+qdy\\ dz=aydx+ \frac{z-xa^2}{ay-a}dy\\ \text{by integrating both sides, we have}\\ z=axy+(z-xa^2)log_e(ay-a)\\ z=axy+blog_e(ay-a)\\ \text{Where a and b are arbitrary constants}$