Answer to Question #143400 in Differential Equations for Mike

Question

Problem:
x^2+y^2=9
How to find answer y''= -x^2+y^2/y^3=-9/y^3 ?( I can't describe the operation type therefore any solution is accepted)

Answer 1

Consider the equation:

$x^2+y^2=9$ (*)

Find derivates of both parts of the equation (*):

$2x+2yy'=0$ (**)

Further, let us find derivates of both parts of the equation (**):

$2+2(y')^2+2yy''=0$ (***)

The equation (**) is equivalent to

$y'=-\frac{x}{y}$ (****)

The equation (***) is equvialent to

$y''=-\frac{1+(y')^2}{y}=$

 | next we use (****) |  

$=-\frac{1+(-\frac{x}{y})^2}{y}=-\frac{y^2+x^2}{y^3}=$

 | next we use (*) |  

$=-\frac{9}{y^3}$