Explanation Given: The second-order equation is y ′′ ( t ) + y ( t ) − y ( t ) 4 = 0 Figure ( 1 ) Approach: Let, d y / d t = v d v / d t = f ( y , v ) ( 1 ) Critical point ( y 0 , v 0 ) for ( 1 ) is given by- v 0 = 0 f ( y 0 , v 0 ) = 0 Use the related phase-plane differential equation to sketch the direction field in the y v -plane- d v d y = f ( y , v ) v ( 2 ) Compare the sketch of direction field with figure ( 1 ) to describe the stability of the critical points. Calculation: Given, y ′′ ( t ) + y ( t ) − y ( t ) 4 = 0 y ′′ ( t ) = − y ( t ) + y ( t ) 4 Compare with equation ( 1 ) d y / d t = v d 2 y / d t 2 = d v / d t = f ( y , v ) = − y + y 4 Solve the above system of equations for critical points

Pearson eText Fundamentals of Differential Equations with Boundary Value Problems -- Instant Access (Pearson+)

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Author: R. Nagle, Edward Saff

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Chapter 5.4, Problem 23E

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All the critical points for the first-order system in the yv-plane by converting the given second-order equation into a first-order system setting v=y′ and sketch the direction fields, describe the stability of the critical points.

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Chapter 5.4, Problem 22E

Chapter 5.4, Problem 24E

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4. Now we'll look at a nonhomogeneous example. The general form for these is y' + p(x)y = f(x). For this problem, we will find solutions of the equation +2xy= xe (a) Identify p(x) and f(x) in the equation above. p(x) = f(x) = (b) The complementary equation is y' + p(x)y = 0. Write the complementary equation. (c) Find a solution for the complementary equation. We'll call this solution y₁. (You only need one particular solution, so you can let k = 0 here.) Y1 = (d) Check that y₁ satisfies the complementary equation, in other words, that y₁+ p(x)y₁ = 0.

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All the critical points for the first-order system in the y v -plane by converting the given second-order equation into a first-order system setting v = y ′ and sketch the direction fields, describe the stability of the critical points.

Solution Summary: The author explains how to find the critical points for the first-order system in the yv -plane.

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To find:

All the critical points for the first-order system in the yv-plane by converting the given second-order equation into a first-order system setting v=y′ and sketch the direction fields, describe the stability of the critical points.

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Chapter 5 Solutions

All the critical points for the first-order system in the y v -plane by converting the given second-order equation into a first-order system setting v = y ′ and sketch the direction fields, describe the stability of the critical points.

Solution Summary: The author explains how to find the critical points for the first-order system in the yv -plane.

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To find: All the critical points for the first-order system in the yv-plane by converting the given second-order equation into a first-order system setting v=y′ and sketch the direction fields, describe the stability of the critical points.

Want to see the full answer?

Chapter 5 Solutions

To find:

All the critical points for the first-order system in the yv-plane by converting the given second-order equation into a first-order system setting v=y′ and sketch the direction fields, describe the stability of the critical points.