Question 1167011
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<pre>

    {{{y*((dy)/(dx))}}} = {{{x(y^4 + 2(y)^2 + 1)}}}  at x= -3 , y=1     (1)


Use the method of separation of variables.


Collect all the term with y and dy on the left side; all the terms with x and dx on the right side


    {{{(y*dy)/(y^4+2y^2+1)}}} = {{{x*dx}}}.


    {{{(1/2)*((d(y^2))/((y^2+1)^2))}}} = {{{(1/2)*(dx^2)}}}


    {{{(d*(y^2+1))/((y^2+1)^2)}}} = {{{(d(x^2))}}}


Take antiderivative; do not forget to introduce an arbitrary constant C


    -{{{1/(y^2+1)}}} = {{{x^2}}} + C       (2)


Find the value of the constant C from initial conditions


    -{{{1/(1^2 + 1)}}} = {{{(-3)^2}}} + C;


    -{{{1/2}}} = 9 + C;  C = - 9 {{{1/2}}} = {{{-19/2}}}.


Substitute  C = {{{-19/2}}} into (2)


    -{{{1/(y^2+1)}}} = {{{x^2}}} - {{{19/2}}}


    -{{{1/(y^2+1)}}} = {{{(2x^2 - 19)/2}}}


    {{{y^2 +1}}} = {{{-2/(2x^2-19)}}}


    {{{y^2}}} = {{{-2/(2x^2-19)}}} - 1 = {{{(-2 - 2x^2 + 19)/(2x^2-19)}}} = {{{-(2x^2-17)/(2x^2-19)}}}


    y = {{{sqrt((17-2x^2)/(2x^2-19))}}}.      <U>ANSWER</U>
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Solved.