Page 100 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 5: Differential Equations

Solution: Notes
2
y 1 x dx x dy 0
1 x dy
or dx 0, Integrating
x 2 y

1 x dx dy a
x 2 y

1
or log x log y a
x

1
or log y log x a
x
y 1
or log a
x x

Example 2: Solve
dy
xy 8x
dx
Solution:

dy
xy 8x 0
dx
dy
or x y 8 0
dx

dy
or xdx 0
y 8
Integrating we have

x 2
log y 8 a
2
Here a being an arbitrary constant.
(B) The Exact Equation
Consider the differential equation
M(x, y) dx + N(x, y) dy = 0 ...(i)
with the condition

M N
...(ii)
y x

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