常微分方程讲义很详细

1、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 4a22a24a23a24a25a24a26a24a27a24a28a24a29a24a30a24a31a24a32 *a33a35a34 f(t; x)a36 (t;x) a37a35a38a35a39a35a40 G a41 k a42a35a43a35a44a35a45a3a35a46 x = (t)a36a35a39a35a47a35a38 I a48a35a49a6a35a7a35a50 (E)a51a35a52a46a53a55a54a57a56 (t)a58a35a59 (E) a60a35a61a35a51a35a62a35a63a35a64a35a65a35a66a46a68a67a42a35a69a35a70 (t) a51a35a71a35a61 k + 1 a72a3a35a73a35a74a49a35a43a35a44a35a51a35a75a33a35a34 f(t; x)a36。

2、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 6a22a24a23a24a25a24a26a24a27a24a28a24a29a24a30a24a31a24a32a34a33a35a24a36 f(t;x)a37 (t;x) a38a40a39a42a41a24a43 G a44a42a45a24a46a24a47a49a48a24a50a24a51a24a52a24a53a24a54a24a55a40a56a58a57a24a59a34a60a49a61a34a62 x2a63a24a64 2.1a65 x5 a66a24a67 5.1, a68a24a69a24a70 ( ; ) 2 G, a71a24a72a24a73 (E) a53a24a54a24a74a24a75a24a57a34a59 x( ) = a41a24a76a24a65a24a77a24a78a24a37a24a47a80a79a24a48a24a81a34a82a24a70a34a。

3、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 2a22a24a23a24a25a24a26a24a27a24a28a24a29a24a30a24a312.1 a32a34a33a34a35a34a36a38a37a34a39a34a36a38a40a34a41a34a42a34a43a44a24a45a47a46a49a48a24a50a24a51a24a52 (E)a. a53a24a54 f(x) a55 Rn a56a24a57a24a58a24a59a24a60a24a61a62 xa63a24a64a24a65a24a66a68a67 Rn a69a71a70a24a72a24a73 . a74a24a75a24a76 x a77a24a78a24a79a24a80a24a65a24a81a83a82a83a84 ta77a24a78a24a85a68a67a86a84(E)a a77a87a78a87a79a87a80a87a65a87a88a8。

4、常微分方程习题2.1 1. xydxdy2= ,并求满足初始条件：x=0,y=1的特解. 解：对原式进行变量分离得 。故它的特解为代入得把即两边同时积分得：eexxycyxxcycyxdxdyy22,11,0,ln,21 2=+=,0)1(.22=+ dyxdxy并求满足初始条件：x=0,y=1的特解. 解：对原式进行变量分离得： 。故特解是时，代入式子得。当时显然也是原方程的解当即时，两边同时积分得；当xycyxyxcycyxydydxxy+=+=+=+=+1ln11,11,001ln1,11ln0,11123 yxydxdyxy321+= 解：原式可化为： xxyxxyxyxyyxyccccxdxxdyyyxydxdy2222222232232)1(1)1)(1(),0(ln1ln21ln1ln2111,0111=+=+=+=+=+）故原。

5、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a5a14a20a21 2a22a24a23a24a25a24a26a24a27a24a28a24a29a30a32a31a32a33a32a34a32a35a32a36a24a37a39a38a41a40a24a42a24a43a24a44a24a45a24a46a24a47a49a48a24a50a24a51a24a52a24a36a24a37a24a53a24a54a49a55a57a56a24a58a24a59a24a60a24a40a24a42a24a61a49a62 6a30a24a63a24a64a24a65a24a66a24a50a24a67a24a68a24a692.1 a70a72a71a72a73a72a74a72a75a72a76a72a77a78a24a79dydx = h(x)g(y) (2.1)a50a32a65a32a66a32a80a32a81a83a82a32a84a32a85a32a86a。

6、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 4 (NH)a20a22a21a22a23a22a20a22a24a22a25a26a28a27a28a29a28a30a28a31a22a32a22a33a35a34a22a36a22a37a35a38a22a39 (NH).a40a28a41a28a42a28a43 (NH) a44a28a45a28a46a28a44a28a47a22a48a22a49a51a50a35a52a22a53a54 (NH)a55 (LH) a56a58a57a59a44a22a60a22a61a22a62a22a63a22a49a65a64a35a66a58a67a69a68a35a70a22a71a35a44a22a72a73a22a74 4.1 (NH)a75a59a76a58a77a59a78a22a79a58a75 (LH) a75a59a76a22a80a22a81a22a82a22a83 (NH) a75a59a76a。

7、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 5a20a22a21a22a23a22a24a22a25a22a26a22a27a22a28a29 x2a30a32a31a34a33a36a35a36a37a36a38a36a39a36a40a36a41a22a42 (NH)a43a36a44a36a45a47a46a32a48a36a31a34a49a47a50a32a39a36a51a22a43a36a52a22a53a29a22a54a36a55a22a56a36a57a58a22a59 x4a43a22a60a22a61a22a31a62a33a22a35a22a63a56a65a64a65a66a65a67a31a68a44a65a45a69a46a70a48a65a43a65a52a65a71a65a72a65a73a22a40a65a41a65a42 (LH) a43a22a74a22a75a22a52a76a47a77a32a78a47a79a32。

8、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 3a22a24a23a24a25a24a26a24a27a24a28a24a29a30 x2a31a33a32a35a34a37a36a30 f(t; x)a38a37a39a37a40a37a41a37a42a37a43a45a44a47a46a24a48a24a49a24a50a24a51a24a52a24a53a55a54a47a56a24a57a59a58a24a60 (E) a49a61a24a62a55a63a47a64a24a65a49a59a66a30a68a67a59a69a59a70a59a71a68a72a74a73a59a75a59a76a50a68a51 f(t; x) a38a24a39a24a32a78a77a24a79a73a59a80a82a81a52a59a83a59a49a68a84a59a85a86 (Peano, 1858-1932)a87a37a88a37a89a65a。

9、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a20a21 8a22a24a23 n a25a24a26a24a27a24a28a24a29a24a30a31a24a32a24a33a24a34a24a35 na36a24a37a24a38a24a39a24a40a24a41a24a42a24a43x(n) = f(t;x;x0; ;x(n 1) (E)na44a24a45a24a46a24a47a24a33a24a48a24a49x1 = x; x2 = x0; ; xn = x(n 1)a50 a35a52a51a24a53a24a54a24a55a24a56a24a57a39a24a58a60a59a24a61a60a62a24a42a60a43a60a63a24a648:x01 = x2;. . . . . . . . . . . . . . . . . . . . .x0n 1 = xn;x0n = f(t;x1; ;xn);a65 (E)n a59a2。

10、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 7a20a22a21a22a23a22a24a22a25a22a26a22a27a22a28a22a29a22a30a22a31a32a25a22a33a34a36a35a36a37a36a38a36a39a36a40a36a41a32a42a22a43a22a44a22a45a22a46a48a47a22a49a22a50a22a51a22a52a22a53a22a54a22a55a57a56a59a58a22a39a32a60a22a61a48a62a36a63a32a64a22a65a22a66a22a67a22a55a22a60a44a22a68a22a46a69a58a22a49a22a34a22a35a22a70a22a71a22a40a32a41a22a42a32a43a22a44a32a45a32a72a22a73a32a74a32a39a22a617.1 a75a77a76a77a78a77a79a。

11、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a6a14a7a19 1a20a22a21a23a25a24a25a26a25a27a29a28a31a30a33a32a22a34a22a35a37a36a22a38a22a39a22a40a37a41a22a42a22a43a22a44a37a26a22a45a22a46a22a47a37a48a22a41a22a49a22a30a33a50a22a51a22a52a37a45a22a42a22a49a22a30a33a53a34a22a54a22a55a22a56a22a57a22a42a22a43a22a44a22a58a37a39a37a59a37a60a37a61a37a62a64a63a37a65a37a30a67a66a37a32a37a34a37a23a22a24a37a26a37a27 x4 a28a69a68a22a70a72a71a69a30a64a73a22a51a22a46a37a47a48a25a41a25a。

12、a0a2a1a4a3a4a5a4a6a4a7a4a8a4a9a4a10a4a11a4a7 a12a14a13a14a15a17a16a14a18a14a19a14a3a14a5a14a6a14a7a20 1a21a23a22a24a23a25a27a26a23a28a23a29a23a30a23a31a23a32a23a33 ,a34a23a35a23a36F(x1; ;xn;u; ux1; ; uxn) = 0; (1.1)a37a39a38 x1; ;xn a36a41a40a43a42a23a44a23a45 ua36 x1; ;xn a35a23a46a23a47a23a48a23a49a23a50a51a53a52a53a54 (x1; ;xn) a55a57a56a59a58a53a60 D a61a59a35a53a48a53a49 u = (x1; ;xn) a62a53a63a53a64a53a65 (1.1)a54a60 D a61a66a35a68a67 , a69a71a70a71a72a54a60 D a61a66a73a71a74a71a。

13、 第一章 初等积分法 第 1讲 微分方程与解 第 2讲 变量可分离方程 第 3讲 齐次微分方程 第 4讲 一阶线性微分方程 第 5讲 全微分方程与积分因子 第 6讲 一阶隐式微分方程 第 7讲 几种可降阶的高阶方程 第 8讲 应用举例 第二章 基本定理 第 09讲 解的存在性与唯一性定理 第 10讲 解的延展 第 11讲 奇解与包络 第 12讲 解对初值的连续依赖性 第三章 线性微分方程组 第 13 讲 一阶微分方程组及 一阶线性微分方程组的一般概念 第 14 讲 线性齐次微分方程组的一般理论 第 15 讲 线性非齐次微分方程组的一般理论 常系数线性微分方程组的解法 (单实。