MHB How can I solve a non-homogeneous equation using substitution?

Joe20
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Hi, I have attached part of my steps for solving the homogeneous equation.
The equation is proven to be homogeneous. However after using substitution of y=zx and its' derivative, I was not able to separate the variables conveniently as shown. Please advise. Thank you!
 

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The difficulty is that this equation is NOT homogenous because of the "+ 1" and "-1" terms.
 

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