Linearizing a Non Linear Differential Eq

[ashketchumall]

Homework Statement

Use a suitable substitution to transform the nonlinear DE (dy)/(dx)+y=y² into a linear equation in the new variable z.
and
without solving the DE, justify the possible methods that can be used to solve the DE found in first part.

Homework Equations

I have no idea what to do in order to solve this.

The Attempt at a Solution

dy/dx=y² -y
dy/dx= y(y-1)

∫dy/y=∫(y-1)dx

ln(y)=y-1??

e^ln(y)=e^(y-1)

y= e^y-1 I don't think this is write, but this is what I have... can someone help?

 

[HallsofIvy]

Did you not understand the problem? Surely you saw the "without solving the DE" part?

(And "$\int y-1 dx$", where y is an unknown function of x, is certainly not "y- 1". If you do not know y as a function of x, you cannot integrate with respect to x. If you wanted to solve the equation, you should instead completely separate "x" and "y":
$$\int \frac{dy}{y(y-1)}= \int dx$$
using "partial fractions" on the left.)

 

[ashketchumall]

Did you not understand the problem? Surely you saw the "without solving the DE" part?

(And "$\int y-1 dx$", where y is an unknown function of x, is certainly not "y- 1". If you do not know y as a function of x, you cannot integrate with respect to x. If you wanted to solve the equation, you should instead completely separate "x" and "y":
$$\int \frac{dy}{y(y-1)}= \int dx$$
using "partial fractions" on the left.)

Sorry I might have written the questions wrong, first part is separate from the second part.
The questions below are the right ones

3a.Use a suitable substitution to transform the nonlinear DE (dy)/(dx)+y=y² into a linear equation in the new variable z.
3b.without solving the DE, justify the possible methods that can be used to solve the DE found in 3a.

 

[CAF123]

Notice your equation is a bernoulli differential equation. There is an explicit method you can use to linearise such equations: See, for example,
http://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx

 

[ashketchumall]

Notice your equation is a bernoulli differential equation. There is an explicit method you can use to linearise such equations: See, for example,
http://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx

You are a life saver! Thankyou so much!

:-)