How do you solve this differential equation #dy/dx=(-x)/y# when #y=3# and #x=4# ?

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Answer 1 · 2018-03-13T02:40:03

# y^2 = 25 - x^2 #

We have:

# dy/dx=(-x)/y# with #y=3# when #x=4#

This is a separable ODE, so we can write:

# y \ dy/dx = -x #

Then we can "separate the variables" :

# int \ y \ dy = - \ int \ x \ dx #

Then we can readily integrate to get:

# 1/2y^2 = - 1/2x^2 + C #

Given the initial condition #y(4)=3# then:

# 1/2 * 9 = - 1/2 * 16 + C => C = 25/2 #

So the Particular Solution is:

# 1/2y^2 = - 1/2x^2 + 25/2 #

# :. y^2 = 25 - x^2 #

Which we note is a circle of radius #5# centred on the origin.