Page 237 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes x (t) = p + q y, y (t) = p q x, z (t) = p(p + q y) + q(p q x)
p (t) = q y + p, q (t) = p x + q ...(5)
Now x (t) = p (t), which gives x = p + , so that t = 0

x = , p = 0, so x = + p ...(6)
similarly y = q 2 ...(7)
Also, it is readily shown that

d
(p + q x) = q y + p + p x + q p q + y
dt
= p + q x

( d p q x )
So = dt
p q x

On integrating we get
log(p + q x) = t + log c
1
or p + q x = c e t ...(8)
1
At t = 0, p = 0, q = 0, x = we get c = +
1
therefore p + q x = + e t ...(9)

Similarly
d
)
(p q y = p + q y + p + q p x p q + x = p + q y
dt
d
)
or (p q y = p + q y ...(10)
dt
On integrating (10) we get
p + q y = 2 e t ...(11)

the constant of integration being 2 .
From (6) and (9) we have
t
q = e p + x
t
t
or q = e + = (e + 1) ...(12)
From (7) we have
t
y = q 2 = (e 1) ...(13)
From (11) we have
t
p = 2 e q + y
t
t
t
= 2 e (e + 1) + (e 1)
t
or p = 2 (e 1) ...(14)

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