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du y 2 3
e  y ,
dy
2
y
3 
du  y e dy ,
2 1 2  y 2 2 1  y 2 2
y
3 
u   y e dy  y e dy   e  1 y  C .
2 2
The last result was got integration by parts.
Consequently, we have
 1  y 2 2  y 2
x  uv    e 1 y   C  e
 2 
or

x  Ce y 2  1 1 y 2 . 
2

2.5 Bernoulli Equation

Equation of kind is named Bernoulli equation

y   P    Qyx  yx n , n  , 0 n  1. (2.36)

If n  0, we will have the considered linear equation
already, and if n  1, will get equation with the separated
variables.
It is easily to show that replacement z  y 1  n allows to take
equation (2.36) to linear one, however more expediently will at
once do replacement Bernoulli y  uv , y  u v  v u  like till

we do it in linear equation.

Example 2.12 To untie equation yx   4 y  x 2 y .

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