Page 301 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes
F F 1 F F 1 F F 1 F F 1
x p p x x p p x
3 3 3 3 4 4 4 4
= 2p p 0 2x p (0) p 4 (0) 0 0 ...(3)
1 3
3 1
The next step is to find F and F such that
2 3
, F F 2 = 0 = F 1 ,F 2 ...(4)
Now
F F F F
= 2x p , 0, 2x p , x
p 3 3 3 1 4
1 p p p
2 3 4
F F F F
= 0, 0, 2p p , p
x 1 3 4
1 x x x
2 3 4
dx 1 dx 2 dx 3 dx 4 dp 1 dp 2 dp 3 dp 4
2x p = 0 2x p p 0 0 2p p p
3 3 3 1 4 1 3 4
p = a ...(4)
2 2
so F = p = a , so F ,F 0 F ,F
2 2 2 2 1 2
dx dp
Also from 3 = 3 , on integration
2x p 2p p
3 1
1 3
F = x p = a ...(5)
3 3 3 3
Again F 1 ,F 3 = 0 , F F 3 0 F 2 ,F 3 0

a a 2x p p a a
p = 2 , p a , p 3 , p 3 3 1 3 2
1 2 2 3 4
2 x 3 x 4 2x 4
so from the relation
du = p dx + p dx + p dx + p dx
1 1 2 2 3 3 4 4
a dx a a a dx
= 2 dx 1 a dx 2 3 3 3 2 4 ...(6)
2
2 x 3 2x 4
On integrating (6) we have the complete integral
a a a
u = 2 x 1 a x a 3 log x 3 2 3 log x 4 a 4 ...(7)
2 2
2 2
Self Assessment

3. Solve for complete integral of
F = p 2 p p x x 2 0
1 2 3 2 3
F = p + p x = 0
1 1 2 2
4. Find the complete integral of
F = x p  x p + p  p = 0
1 1 2 2 3 4
F = p + p  x  x = 0
1 1 2 1 2

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