# Use u = y' substitution to solve (y + 1) y'' = (y')^2

• s3a
In summary, the student is struggling with understanding the substitution u = y' and how to apply the chain rule to solve the differential equation. They are confident in their ability to solve the rest of the problem.
s3a

## Homework Statement

Solve the following differential equation, by using the substitution u = y'.:
(y + 1) y'' = (y')^2

## Homework Equations

I'm assuming: Chain Rule

## The Attempt at a Solution

My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would appreciate it if someone could show me how!

Other than that, I should be fine.

Re-write:

(y+1) u' = u²

then rearrange to get u's on the left and y's on the right and integrate.
(this is a classical manipulation in classical mechanics)

s3a said:

## Homework Statement

Solve the following differential equation, by using the substitution u = y'.:
(y + 1) y'' = (y')^2

## Homework Equations

I'm assuming: Chain Rule

## The Attempt at a Solution

My problem is, simply, that I don't get how to go from u = y' to y'' = u du/dy, and I would appreciate it if someone could show me how!

Other than that, I should be fine.

I will use prime for differentiation with respect to the independent variable which I will assume to be ##x##. We have the substitution ##u(y) = y'##. Differentiating both sides with respect to ##x## gives$$y'' = u'(y) = \frac{du}{dy}\cdot y' = u \frac{du}{dy}$$It's just the chain rule and I switched the factors in the last step.

## 1. What is the purpose of using the u = y' substitution in this equation?

The u = y' substitution is used to simplify the equation and make it easier to solve. By substituting u for y', the second derivative (y'') term is eliminated, leaving a first-order differential equation that can be solved using standard methods.

## 2. How do you perform the u = y' substitution in this equation?

To perform the u = y' substitution, simply replace every instance of y' in the equation with u. This includes the y'' term, which becomes u' after the substitution. The resulting equation will be in terms of u instead of y'.

## 3. Can you explain why the second derivative term (y'') is eliminated after performing the u = y' substitution?

When u = y' is substituted in the equation, the derivative of u with respect to x becomes u'. This eliminates the second derivative term (y'') and reduces the equation to a first-order differential equation.

## 4. Are there any restrictions or limitations when using the u = y' substitution?

Yes, there are some restrictions and limitations when using the u = y' substitution. It is only applicable when the equation is in the form of (y + 1) y'' = (y')^2. It cannot be used for other types of equations, and it may not always lead to a simpler equation.

## 5. Are there any other methods for solving this type of equation besides using the u = y' substitution?

Yes, there are other methods for solving this type of equation, such as separation of variables or using an integrating factor. However, the u = y' substitution is a common and effective method for simplifying and solving this specific type of equation.

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