Question

Using Charpit’s method, find a complete integral of the following equations: xpq + yq^2 = 1

Answer 1

To find a complete integral of $f=xpq+yq^2-1=0$

$\implies$ The auxiliary equations are $\frac{dx}{xq}=\frac{dy}{2qy}=\frac{dz}{xpq+2yq^2}=\frac{dp}{-pq}=\frac{dq}{-q^2}$. Hence $\frac{dp}{-pq}=\frac{dq}{-q^2}$ gives p-aq. Thus g=p-aq=0. Solving f=0 and g=0 for p and q, we get $q=\frac{1}{\sqrt{ax+y}}, p=\frac{q}{\sqrt{ax+y}}$. Hence $dz=\frac{adx+dy}{\sqrt{ax+y}}$. Thus $(z+b^2)=4(ax+y)$ is the required complete integral.