Simple Seperable Variables Question

[Tracker]

Solve the following differential equation by separation of variables:

 

[math] \frac{xdy}{dx} = 4y [/math]

 

My solution is:

 

[math]\int \frac{dy}{4y} = \int \frac{dx}{x} \Rightarrow ln{\mid 4y \mid} = ln{\mid x \mid} + c [/math]

 

The book's solution is:

 

[math] y = cx^4 [/math]

 

Can anyone show me how they came up this this solution?

 

Thank you.

[DrRocket]

Solve the following differential equation by separation of variables:

 

[math] \frac{xdy}{dx} = 4y [/math]

 

My solution is:

 

[math]\int \frac{dy}{4y} = \int \frac{dx}{x} \Rightarrow ln{\mid 4y \mid} = ln{\mid x \mid} + c [/math]

 

The book's solution is:

 

[math] y = cx^4 [/math]

 

Can anyone show me how they came up this this solution?

 

Thank you.

 

Take the log of both sides of the solution in the book.

[Tracker]

Take the log of both sides of the solution in the book.

 

[math] y = cx^4 \Rightarrow ln{y} = ln{cx^4} \rightarrow ln{y} = ln{c} + 4ln{x} \rightarrow ln{y} = 4ln{x} + c [/math]

 

I don't understand how the four gets inside the ln to become [math] ln{4y} [/math]

 

Thank you for the help.

 

Cheers.

[DrRocket]

[math]\int \frac{dy}{4y} = \int \frac{dx}{x} \Rightarrow \frac{1}{4} ln{\mid y \mid} = ln{\mid x \mid} + c [/math]