By Qingkai Kong

This article is a rigorous remedy of the fundamental qualitative idea of normal differential equations, at the start graduate point. Designed as a versatile one-semester path yet supplying adequate fabric for 2 semesters, a brief path covers center issues corresponding to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré—Bendixson theorem, and bifurcation conception, and second-order subject matters together with oscillation thought, boundary price difficulties, and Sturm—Liouville difficulties. The presentation is obvious and easy-to-understand, with figures and copious examples illustrating the that means of and motivation in the back of definitions, hypotheses, and basic theorems. A thoughtfully conceived choice of workouts including solutions and tricks toughen the reader's knowing of the fabric. necessities are constrained to complex calculus and the undemanding concept of differential equations and linear algebra, making the textual content compatible for senior undergraduates in addition.

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6) V ( X , ~ S, , t)=~js~-2tx~s+t~~=O. Here t h e e l l i p s e ANMC a c t s a s o r d i n a t r i x of t h e family of p a r a b o l a s , s i n c e i n each of i t s p o i n t s N t h e p a r a b o l a p a s s i n g through N i s w e l l d e f i n e d . Johann B e r n o u l l i ' s d e t e r m i n a t i o n of t h e s a f e t y p a r a b o l a family of parabolas j u s t d e f i n e d - - b e i n g t h e envelope o f t h e proceeded a l o n g t h e f o l l o w i n g l i n e s : D i f f e r e n t i a t e e q u a t i o n ( 2 .

E . c u r v e s BB'B" d e f i n e d by areaABD= =areaAB'D'; s e e f i g u r e 10) i n a f a m i l y of e l l i p s e s over t h e same axis, and f o r (b) equaZ ares trajectom'es ( i . e . c u r v e s BB'B" d e f i n e d by arc AE=arc A % ' ) 31 The bractiystochrone and its aftermath i n any f a m i l y of (what he c a l l e d ) "curves of t h e same s o r t " . f i g . 10 Johann B e r n o u l l i d i d n o t c l a r i f y t h e meaning of t h e term "curves of t h e same s o r t " when he p u b l i s h e d t h e s e problems; i n f a c t - by mentioning a f a m i l y of s i m i l a r p a r a b o l a s a s s p e c i f i c example - he s u g g e s t e d t h a t h e had f a m i l i e s of s i m i l a r curves i n mind.

10) must have a double r o o t i n s on t h e envelope. S t r a i g h t f o r w a r d c a l c u l a t i o n of t h i s double r o o t then y i e l d s e q u a t i o n ( 2 . 9 ) . 5 Conclusion L e i b n i z ' s new a p p l i c a t i o n of t h e d i f f e r e n t i a l c a l c u l u s was a remarkable achievement, i n t h a t i t demonstrated t h a t t h e c a l c u l u s was a p p l i c a b l e n o t o n l y t o a s i n g l e c u r v e , b u t a l s o t o f a m i l i e s of c u r v e s . "" However, t h e envelope a r t i c l e s 1692 and 1694 o n l y c o n s t i t u t e an i s o l a t e d e p i s o d e i n t h e development of p a r t i a l d i f f e r e n t i a t i o n .