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(a) Solve the differential equation $ y' = 2x \sqrt {1 - y^2}. $(b) Solve the initial-value problem $ y' = 2x \sqrt {1 - y^2}, y(0) = 0, $ and graph the solution.(c) Does the initial-value problem $ y' = 2x \sqrt {1 - y^2}, y(0) = 2, $ have a solution? Explain.

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Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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$$\begin{array}{l}{\text {…

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(a) Solve the differential…

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Solve the given differenti…

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Find the particular soluti…

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Solve the differential equ…

We have part where we have had, as equals 2 x under root 1 minus y squared. Therefore, it is the question d y over under root 1 minus y squared equal to equation 2. It point, therefore, it is sine inverse y equal to 2 into x, squared over 2 plus c, so this cancels his power. Therefore, we have sine inverse y equals h. Squared plus c. We have part b, that is the given parameter, is y 0 equals 0 and therefore we have sine inverse 0 equals 0 plus c. Therefore, it is 0 equals c and now sine inverse y equals x squared, therefore, we have y equals sine x square. This is the answer part b of the problem. Now we have part c of the problem where sine inverse 2 is equal to 0 plus c or it is sine inverse y is equal to x, squared plus sine inverse 2. So since sine inverse 2 as a non real result, c is undefined. That is c is undefined in x, y plane and since c is undefined. Therefore, y x has not a particular solution at y equal to to the given prone.

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