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Problem

Solve the equation $ e^{-y}y' + \cos x = 0 $ and …

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Problem 23 Easy Difficulty

(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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Related Courses

Calculus 2 / BC
Calculus: Early Transcendentals

Chapter 9

Differential Equations

Section 3

Separable Equations

Related Topics

Differential Equations

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Top Calculus 2 / BC Educators
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Lectures

Video Thumbnail

13:37
Differential Equations - Overview

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.
Video Thumbnail

33:32
Differential Equations - Example 1

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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Video Transcript

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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Top Calculus 2 / BC Educators
Heather Zimmers

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Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Joseph Lentino

Boston College

Calculus 2 / BC Courses

Lectures

Video Thumbnail

13:37
Differential Equations - Overview

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.
Video Thumbnail

33:32
Differential Equations - Example 1

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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