CHARPITS METHOD

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CHARPITS METHOD
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Charpits method This is a general method for
finding the solution of a non-linear partial
differential equation Consider the equation f
(x, y, z, p, q) 0 .(1) Since z depend on x
and y, we can write
(2)
Now if we can find another relation involving
x,y,z,p,q such as ?(x,y,z,p,q)0 .(3) then
we can solve (1) and (3) for p and q and
substitute in (2). This will give the solution
provided (2) is integrable.
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Subsidiary equations are
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(1) Solve ( p2q2 )y qz Let f(x,y,z,p,q)(
p2q2 )y-qz0
The last two of these give p dp q dq 0 (i)
Integrating, p2q2c2 (ii) Now to
solve (i) and (ii), put p2q2c2 in (i), so that
qc2 y/z Substituting this value of q in
(ii), p
c?(z2-c2y2)/z
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Hence
or z dz c2y dy c?(z2-c2y2)dx
Integrating, we get ??(z2-c2y2)c x a or z2
(acx)2c2y2 which is the required complete
integral .(Solution)
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(2) px qy pq
f(x,y,z,p,q) ? px qy-pq0
Subsidiary equations are
first two terms
(p/q)a,or paq, where a is a constant

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Using (p /q)a this in the given equation px q
y p q we get (a q) x q yaq2 or
or a z (axy)2 2b
This is a complete solution of the given equation.
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(3) (pq) (pxqy) 1

f(x,y,z,p,q) ? (pq) (pxqy)-10 .(i)
? (p2xq2ypqxpqy)-10

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(p/q)a in the given equation (pq) (px q y)
1 we get (a q q) (a q) x q y1
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(4) p x y p q q y y z
f(x,y,z,p,q) p x y p q q y - y z0
Substituting this value of p in the given equation
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a x y a q q y y z0 q y (z- ax)/ (a y)
d z p d x q d y a dx y (z- ax)/ (a
y)dy (dz -a dx )/ (z- ax) y/ (a y) d
y integrating log (z- ax) y a log (a y)
b This is a complete solution of the given
equation.
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(5) z2 p qxy
f(x,y,z,p,q) ? z2 p q x y 0 ...(i)

Integrating (p x /q y )constant
px a2qy a arbitrary
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Substituting value of p in the equation z2 p
qxy we get q z/ya and p az/x dz pdx
qdy (az/x )dx (z/ya )dy dz/z (a/x
)dx (1/ya) dy integrating log z (a logx)
(1/a) log y log b OR z bxay(1/a) is
the required solution
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(6) p2 xq2y z
f(x,y,z,p,q) ? p2 xq2y z 0 .(i)

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