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Solve the following equation using charpits method

Screenshot_12

Solution: Given that,

Screenshot_21

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

Screenshot_14

Screenshot_15

Taking 1st and 4th ratio, we get

Screenshot_16⇒dp=6xdx
Integrating we get,
p = 3x^2+a
Again taking 2nd and 5th ratio, we get
Screenshot_17⇒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
Screenshot_19Which is the required solution.

 

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