Read the summary and the most important questions on Differential Equations & Linear Algebra | 9781292025131 | Stephen Goode, Scott Annin

• 1 First-Order Differential Equations

• 1.1 How Differential Equations Arrise

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• What makes a differential equation?

  Because this mathematical statement involves derivatives of the unknown function y(t).

• How many solutions does the function have?

  This differential equation has an infinite number of solutions parameterized by the constants c₁ and c₂.

• What are Orthogonal trajectories?

  Orthogonal trajectories are when all curves of family F intersect the curves of family G at right angles.

• 1.2 Basic Ideas and Terminology

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• What is the difference between an ordinary and a partial differential equation?

  In an ordinary differential equation the unknown function y(x) is only dependent on x.
  In a partial differential equation the unknown function f(y,x) is dependent on both y and x.

• What is the order of a differential equation?

  The order of a differential equation is the order of the highest derivative occurring in a differential equation.

• When is a solution called a general solution?

  When a solution to an nth-order differential equation on an interval I satisfies the conditions:
  1. The solution contains n constants.
  2. All solutions to the differential equation can be obtained by assigning appropriate values to these constants.

• When is a solution called particular?

  When the solution does not contain any arbitrary constants.

• What is an initial-value problem?

  An nth-order differential equation with n auxiliary conditions of the form:
   , , 
  where y₀, y₁,..., y_n-1 are  constants.

• 1.3 The Geometry of First-Order Differential Equations

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• What is a solution curve?

  A graphical representation of one solution to a differential equation.

• What is meant with the existence of a solution?

  Does the inital-value problem have any solutions?