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Differential Equations

Short Answer Type

111. Solve the following differential equation:

open parentheses straight x squared straight y plus straight x squared close parentheses space dx space plus space left parenthesis straight x space straight y squared minus straight y squared right parenthesis space dy space equals space 0

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112. Solve the following differential equation:

straight y left parenthesis 1 minus straight x squared right parenthesis dy over dx space equals space straight x left parenthesis 1 plus straight y squared right parenthesis

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113. Find the general solution of each of the following equations
$dy over dx equals straight x to the power of 5 plus straight x squared minus 2 over straight x$

The given differential equation is
$dy over dx space equals space straight x to the power of 5 plus straight x squared minus 2 over straight x$
Separating the variables, we get, $dy space equals space open parentheses straight x to the power of 5 plus straight x squared minus 2 over straight x close parentheses dx$
Integrating, we get, $straight y equals straight x to the power of 6 over 3 plus straight x cubed over 3 minus 2 space log space open vertical bar straight x close vertical bar space plus space straight c$
where c is an arbitrary constant.