Question 389723
find an expression for {{{(d^2y)/(dx^2)}}} if {{{2x^2 =3y^2+9}}}. 
Write your answer in terms of y( no x's) the answer 
should be in simplest form.
<pre>
{{{2x^2 =3y^2+9}}} 
{{{4x =6y*expr(dy/dx)+0}}}
{{{4x =6y*expr(dy/dx)}}}
{{{2x =3y*expr(dy/dx)}}}
{{{(2x)/(3y)=expr(dy/dx)}}}
{{{dy/dx=(2x)/(3y)}}}
{{{(d^2y)/(dx^2)=(3y*2-2x*3expr(dy/dx))/((3y)^2)}}}

Simplify, substituting {{{(2x)/(3y)}}} for {{{(dy)/(dx)}}}

{{{(d^2y)/(dx^2)=(6y-2x*3*expr((2x)/(3y)))/(9y^2)}}}

{{{(d^2y)/(dx^2)=(6y-4x^2)/y}}}

Solve the original equation for {{{x^2}}}

{{{2x^2 =3y^2+9}}}

{{{x^2 =(3y^2+9)/2}}}

Substitute that in {{{(d^2y)/(dx^2)=(6y-4x^2)/y}}}

{{{(d^2y)/(dx^2)=(6y-4*((3y^2+9)/2))/y}}}

Cancel the 2 into the 4:

{{{(d^2y)/(dx^2)=(6y-2*(3y^2+9))/y}}}

{{{(d^2y)/(dx^2)=(6y-6y^2-18)/y}}}

There are several different ways to rewrite that if you like:

{{{(d^2y)/(dx^2)=6(y-y^2-3)/y}}}

or

{{{(d^2y)/(dx^2)=6-6y-18/y}}}

or 

{{{(d^2y)/(dx^2)=expr(-6/y)*(y^2-y+3)}}}
 
Edwin</pre>