Solution: Given that,

Let, f(x, y, z, p, q) = p – 3x^2 – q^2 –y

Now, fx = – 6x

fy= 1

fz= 0

fp=1

fq= – 2q

We know that the charpitals auxiliary system is

Taking 1st and 4th ratio, we get

⇒dp=6xdx

Integrating we get,

p = 3x^2+a

Again taking 2nd and 5th ratio, we get

⇒2qdq=dy

⇒q^2=y +b

⇒ q=√(y+b)

Finally, we get

dz = pdx+qdy

⇒dz = (3×2+a)dx+√(y+b)dy

Integrating it we get

Which is the required solution.