Page 283 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
dp dq dx dy
or
2p 2y 2q 2x 2x 2p 2y 2q
dp dq dx dy
or
2(p q x ) y 2(p q x ) y
or p + q = x + y + c

or (p x) + (q y) = c ...(1)
Also the given equation can be written as
2
2
(p x) + (q y) = (x y) 2 ...(2)
Putting the value of (p x) from (1) in (2)
2
2
{c (q y)} + (q y) = (x y) 2
2
2
2
or 2(q y) 2c (q y) + c (x y) = 0
2
2c [4c 2 8{c 2 (x y ) }]
q y =
2 2
c 1 2 2
= x ) y c },
2 2
1 2 2
q = y [c x ) y c }]
2
p x = c (q y)

1
= c [c x ) y 2 c 2 }]
2

1 2 2
p = x {c x ) y c }]
2
Also we know that dz = p dx + q dy.

1 2 2 1 2 2
= [x {c x ) y c }]dx [y {c x ) y c ]}]dy
2 2
c dx c dy 1 2 2
= x dx y dy [ 2(x y ) c } {dx dy }
2 2 2

x 2 y 2 cx cy 1 2 2 dt 2 2
Z = (t c ) if 2(x y ) t
2 2 2 2 2 2

1 t c 2
2 2
or 2Z = x 2 y 2 cx cy (t c ) log{t t (t 2 c 2 )} k
2 2 2

Example 10: Solve by Charpit s method:
pxy + pq + qy = yz.

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