SOLUTION: solve the differential equation: y dy/dx = x(y^4 + 2(y)^2 + 1) at x= -3 , y=1

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Question 1167011: solve the differential equation:
y dy/dx = x(y^4 + 2(y)^2 + 1) at x= -3 , y=1
Found 2 solutions by solver91311, ikleyn:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!




Initial condition:







or

No solution for .











John

My calculator said it, I believe it, that settles it


I > Ø

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

    y%2A%28%28dy%29%2F%28dx%29%29 = x%28y%5E4+%2B+2%28y%29%5E2+%2B+1%29  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


    %28y%2Ady%29%2F%28y%5E4%2B2y%5E2%2B1%29 = x%2Adx.


    %281%2F2%29%2A%28%28d%28y%5E2%29%29%2F%28%28y%5E2%2B1%29%5E2%29%29 = %281%2F2%29%2A%28dx%5E2%29


    %28d%2A%28y%5E2%2B1%29%29%2F%28%28y%5E2%2B1%29%5E2%29 = %28d%28x%5E2%29%29


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


    -1%2F%28y%5E2%2B1%29 = x%5E2 + C       (2)


Find the value of the constant C from initial conditions


    -1%2F%281%5E2+%2B+1%29 = %28-3%29%5E2 + C;


    -1%2F2 = 9 + C;  C = - 9 1%2F2 = -19%2F2.


Substitute  C = -19%2F2 into (2)


    -1%2F%28y%5E2%2B1%29 = x%5E2 - 19%2F2


    -1%2F%28y%5E2%2B1%29 = %282x%5E2+-+19%29%2F2


    y%5E2+%2B1 = -2%2F%282x%5E2-19%29


    y%5E2 = -2%2F%282x%5E2-19%29 - 1 = %28-2+-+2x%5E2+%2B+19%29%2F%282x%5E2-19%29 = -%282x%5E2-17%29%2F%282x%5E2-19%29


    y = sqrt%28%2817-2x%5E2%29%2F%282x%5E2-19%29%29.      ANSWER

Solved.