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(15 points) A Bernoulli differential equation is one of the form

`(dy)/(dx)+P(x)y=Q(x)y^(n)`

Observe that, if

`n=0`

or 1 , the Bernoulli equation is linear. For other values of

`n`

, the substitution

`u=y^(1-n)`

transforms the Bernoulli equation into the linear equation

`(du)/(dx)+(1-n)P(x)u=(1-n)Q(x)`

Consider the initial value problem

`xy^(')+y=-5xy^(2),y(1)=-2.`

(a) This differential equation can be written in the form

`(**)`

with

```
P(x)=
Q(x)=
n=
```

(b) The substitution

`u=`

`?`

will transform it into the linear equation

`(du)/(dx)+`

`?`

`u=`

`?`

(c) Using the substitution in part (b), we rewrite the initial condition in terms of

`x`

and

`u`

:

`u(1)=`

`?`

(d) Now solve the linear equation in part (b), and find the solution that satisfies the initial condition in part (c).

`u(x)=`

`?`