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

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Answer 1 · 2018-01-06T20:13:40

# y^2 = 2-x^2 #

We have:

# y'=-x/y #

This is a first order linear separable ordinary differential equation. If we collect terms we have:

# ydy/dx=-x #

We can now "separate the variables and integrate:

# int \ y \ dy = int \ -x \ dx #
# :. 1/2y^2 = -1/2x^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:

# 1/2y^2 = -1/2x^2 + 1#
# :. y^2 = -x^2 + 2#

Hence:

# y^2 = 2-x^2 #