By Earl A. Coddington
"Written in an admirably cleancut and within your budget style." — Mathematical Review. an intensive, systematic first path in undemanding differential equations for undergraduates in arithmetic and technological know-how, requiring merely easy calculus for a heritage, and together with many routines designed to advance students' approach in fixing equations. With difficulties and solutions. Index.
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Extra info for An Introduction to Ordinary Differential Equations (Dover Books on Mathematics)
C) SOLUTION. Rewrite this as By the linear independence of and give . , set These are solvable, Example 3. (Basis) Consider (1) in the text. Let the 's be continuous on I and suppose is in I. Let denote the solution satisfying initial conditions , and let denote the solution satisfying initial conditions . Is a basis for the DE (1) on I? SOLUTION. 4(c) for that: , so are LI and the set is a basis (that is, a basis of solutions, or a fundamental set) for (1). Example 4. (Basis) Show whether or not and on the interval comprise a basis for the DE .
Check to see if (A) is exact. Thus, its left side is so (A) is exact. , so , Thus, constant and the general solution, in implicit form, is (B) If you want to check (B), take a differential of it: That step gives which is the given equation, (A). , Example 2. (Exact equations) Solve (A) SOLUTION. We can check for exactness, finding that (A) is indeed exact and then proceeding as in Example 1, above. However, look, there are no 's in the first term and no 's in the second, so just simply integrate: so the general solution is , (B) in implicit form.
However, in applications-oriented sections, like this one, each application tends to be unique, so working one problem will probably not help much in working another. Thus, the examples we will give in this section of the solution manual will be few. EXAMPLES Example 1. (Qualitative phase line analysis) Consider the autonomous equation . (a) Obtain the phase line, and from it determine any equilibrium points and their stability. 1 to determine any equilibrium points and their stability. SOLUTION.
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