Differential Equations Verify that x cy 1, where c is an arbitrary nonzero const

Question

Differential Equations

Verify that x cy 1, where c is an arbitrary nonzero constant, is a one-parameter family of implicit solutions to the following differential equation, and plot several of the solution curves using the same coordinate axis. dy STEP 1: Use implicit differentiation to obtain an expression for dy/dx in terms of both x and y. dy dx STEP 3: Substitute this expression into the differential equation and verify that the equation is satisfied for a (x, y) such that x cy 3 1. STEP 3: Graph several of the solution curves on the same set of axes. Select the correct graph.

Explanation / Answer

2x+2cydy/dx = 0

dy/dx = -x/cy

given dy/dx = xy / x2 - 1

then xy / x2 - 1 = -x/cy

cy2 = -x2 + 1

c = (-x2 + 1) / y2

graph 2 is correct as the value of y changes and radius 1 is constant.

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