An introduction to ordinary differential equations"Written in an admirably cleancut and economical style." — Mathematical Review. A thorough, systematic first course in elementary differential equations for undergraduates in mathematics and science, requiring only basic calculus for a background, and including many exercises designed to develop students' technique in solving equations. With problems and answers. Index. 
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Good for concise definitions and information, but not an easy read. It's just... OLD.
Review: An Introduction to Ordinary Differential Equations
User Review  Dustin Tran  GoodreadsMath 204  Ordinary Differential Equations  Jon Wilkening Random chapters. Old, ugly typeset font and archaic notation. I'm not aware of any better textbooks, but I'm not a fan of this text. It could just be a biased review coming from an algebraist, but the notation becomes too cumbersome for me. Read full review
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annihilator method Bessel equation c2 any constants calculus Chap characteristic polynomial complex number Compute Consider the equation constant coefficients continuous function denoted derivatives determinant differentiable function equation L(y equation of order Euler equation example EXERCISES f satisfies fa(x fc(x Find all solutions following equations given hence Hint homogeneous equation implies indicial polynomial initial conditions initial value problem integral interval I containing Legendre equation linear combination linear equations linearly independent solutions Lipschitz condition Lipschitz constant multiplicity nonhomogeneous equation nontrivial solution obtain particular solution positive constant positive integer power series power series expansions proof of Theorem Prove real constants real number realvalued functions regular singular point result satisfies a Lipschitz series converges Show solution of L(y solution satisfying solve successive approximations Suppose tion unique solution valid vector W(fa Wronskian zero