How do I solve the nonlinear differential equation y'=−x/y under initial condition y(1)=1?

Conrad Beltran 2022-09-20 Answered
How do I solve the nonlinear differential equation y = x y under initial condition y(1)=1?
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Answers (1)

xjiaminhoxy4
Answered 2022-09-21 Author has 9 answers
This is a first order linear separable ordinary differential equation. If we collect terms we have:

y d y d x = - x

We can now "separate the variables and integrate:

  y   d y =   - x   d x
1 2 y 2 = - 1 2 x 2 + C

Using the initial condition y(1)=1 we find:

1 2 = - 1 2 + C = 1 C = 1

Thus we have the particular solution:

1 2 y 2 = - 1 2 x 2 + 1
y 2 = - x 2 + 2

Hence:

y 2 = 2 - x 2
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