# New PDF release: A First Course in Partial Differential Equations: with

By H. F. Weinberger

Textual content provides the overall houses of partial differential equations comparable to features, domain names of independence, and greatest rules. suggestions.

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Additional resources for A First Course in Partial Differential Equations: with Complex Variables and Transform Methods (Dover Books on Mathematics)

Example text

2 9. 0/ D a has a unique solution. 10. 0/ D a > 0, is not deﬁned for all t 0. 11. 0/ D a > 0, is deﬁned for all t 0. 12. Show that the solutions of x 0 D sin x are deﬁned on all t 2 R. 13. Show that the solutions of x 0 D arctan x are deﬁned on all t 2 R. 14. 1 C x 2 / are deﬁned on all t 2 R. 15. 0/ D 1, has a unique solution deﬁned for all t and ﬁnd it. 16. 0/ D for all t and ﬁnd it. 1, has a unique solution deﬁned 17. 0/ D 0 is odd. 18. x/ > 0, resp. x/ cannot be even. 38 2 Theory of ﬁrst order differential equations 19.

Y/ above. x; y/ be two arbitrary points in the rectangle . 13) x0 y0 which corresponds to integrating the differential form ! along the path . , that is, Fx D M; Fy D N . x; s/ds D @x y0 y0 @x 50 3 First order nonlinear differential equations y 2 =( 1 , 2 ) ( 1 , 2 ) (x,y) (x,y0 ) (x0 ,y0 ) 1 O 2 x 1 Fig. 4. x; y/: To prove that Fy D N , we notice that the ﬁrst integral is a function of x only. 8. x C 2y/dy D 0. x C y / for all x; y and hence c 0. If c > 0, this is a family of ellipses centered at the origin.

It remains to prove the uniqueness. 12) coincide therein. t/j > 0, we divide by A ﬁnding 1 Ä Lı, a contradiction because we have chosen ı such that Lı < 1. t/ on the interval jt t0 j Ä ı. t0 ˙ ı/. We can now repeat the procedure in the interval Œt0 C ı; t0 C 2ı and 36 2 Theory of ﬁrst order differential equations Œt0 2ı; t0 ı. t/ for all t 2 Œt0 2ı; t0 C 2ı. t/ for all t 2 Œa; b. This completes the proof. t0 ; x0/. t; x/ 2 W jt t0 j Ä r; jx x0 j Ä r º, for some r > 0. 5 proved in the preceding section.