Page 286 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 18: Charpit s Method for Solving Partial Differential Equations

Putting this value of p in (1), Notes

a qy (a qy ) 2 2
2
2
(x y ) q xy 2 q 1 = 0
x x
a qy
or {(x 2 y 2 )q (a qy ) } xyq 2 1 = 0
y
x
a qy 2 2
or (x q ay ) xyq 1 = 0
x
2
2
2
or (a qy) (x q ay) + x yq x = 0
2
2
2
or aq (x + y ) = a y + x.
2
a y x
q = 2 2
( a x y )
2
2
1 (a y x )y a x y
and p a
x ( a x 2 y 2 ) ( a x 2 y 2 )
Putting values of p and q in dz = p dx + q dy, we get
2
2
(a x y )dx (a y x ).dy
dz = 2 2
( a x y )
(x dx y dy ) x dy y dx
or dz = a
x 2 y 2 ( a x 2 y 2 )
Integrating,

a 2 2 1 1 y
z log(x y ) tan . b
2 a x
Self Assessment

Apply Charpit s method to find the complete integrals of:
2
1. pxy + qp + qy = y .
2
2. q = 3p .
2
2
3. p 3x = q y.
2
2
4. z = px + qy + p + q .
2
2
5. 2 (pq + py + qx) + x + y = 0.
2
6. Zxp q = 0
18.3 Special Types of First Order Equations
In the section we shall consider some special types of first-order partial differential equations
whose solutions may be obtained easily by Charpit s Method.

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