定義
![齊次微分方程](/img/3/879/wZwpmLwcjN5ATO0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
形如 的一階微分方程稱為齊次微分方程,簡稱微分方程。
方程特點
![齊次微分方程](/img/8/2a1/wZwpmL3YTN4MDO0QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL0YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
齊次微分方程的特點是其右端項是以為變元的連續函式。
![齊次微分方程](/img/6/dd2/wZwpmLyQDN4AzMxQzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLxIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/8/78a/wZwpmLwATM0UDM1IDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL4QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![齊次微分方程](/img/1/cd3/wZwpmL2ETN5gzN3EjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLxIzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
例如, 是齊次微分方程,它可以轉化為:,即。
方程的解
齊次微分方程通過變數代換,可化為可分離變數微分方程來求解。
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/d/5bb/wZwpmL2gjM3QDM4QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
令或,
![齊次微分方程](/img/f/093/wZwpmL2QzN0EDOxQzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL3AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次微分方程](/img/d/5bb/wZwpmL2gjM3QDM4QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/84f/wZwpmL1ITNxQTMxYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
其中是新的未知函式,對兩邊求導,則有:,
![齊次微分方程](/img/3/879/wZwpmLwcjN5ATO0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/1/15c/wZwpmLwcDOzgjN2QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLxczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
將其代入,得:,
![齊次微分方程](/img/b/f3c/wZwpmL4MDMxADN2UzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL1UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
分離變數,得:
![齊次微分方程](/img/c/999/wZwpmLyETMwczM2YTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2EzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
兩邊積分,得:,
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/879/wZwpmLwcjN5ATO0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
求出積分後,再將回代,便得到方程的通解。
求解步驟
![齊次微分方程](/img/d/5bb/wZwpmL2gjM3QDM4QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(1)作變換,將齊次方程轉化為分離變數的微分方程;
(2)求解可分離變數的微分方程;
![齊次微分方程](/img/8/2a1/wZwpmL3YTN4MDO0QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL0YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/a/ed1/wZwpmL4ITO2IzMyQTM5czN0UTMyITNykTO0EDMwAjMwUzL0EzLwEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(3)用代替步驟(2)中所求通解中的(即變數還原),就可以得到原方程的通解。
注意事項
![齊次微分方程](/img/e/708/wZwpmLzEDO5IzM0QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次微分方程](/img/9/348/wZwpmLyATN4ATNwUTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL4YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次微分方程](/img/d/83f/wZwpmL2gzN0cDMxMjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzL1YzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/1/15c/wZwpmLwcDOzgjN2QzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0MzLxczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/6/a66/wZwpmLyIDO1QTNzQTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL0gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/879/wZwpmLwcjN5ATO0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/6/8b0/wZwpmL3EDO3gjM3YzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL2EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/879/wZwpmLwcjN5ATO0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次微分方程](/img/4/20c/wZwpmLyADM1YjM5AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL2IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
如果有,使得,則顯然也是方程的解,從而也是方程的解;如果,則方程變成,這是一個可分離變數微分方程。
典例
例1
![齊次微分方程](/img/8/7e0/wZwpmLxMTNxYDM1MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLxEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
求解方程。
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/d/5bb/wZwpmL2gjM3QDM4QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/84f/wZwpmL1ITNxQTMxYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
解:令,則,,
![齊次微分方程](/img/9/0cd/wZwpmL1UDM4UzM5czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzL1AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/b/a28/wZwpmL2UTN1YDO1gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLyUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
原方程變為:,即;
![齊次微分方程](/img/b/c57/wZwpmL4cDNzMDN3gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzLxQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
分離變數可得:,
![齊次微分方程](/img/0/dd9/wZwpmL2UzMyMDN0cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
左右兩端同時積分可得:,
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/8/0b1/wZwpmL4QzM4kjN2UzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL0MzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
將 代入,便可得到原方程的通解為:,其中 C 為任意常數。
例2
![齊次微分方程](/img/3/ed5/wZwpmL0gDMycjM4cTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzL4YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
求方程的通解。
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/d/5bb/wZwpmL2gjM3QDM4QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次微分方程](/img/3/84f/wZwpmL1ITNxQTMxYzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL2MzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
解:令,則,,
![齊次微分方程](/img/f/c92/wZwpmLyEzNxQzN1gzM2EzM1UTM1QDN5MjM5ADMwAjMwUzL4MzL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次微分方程](/img/8/4a7/wZwpmL1IzNygjM0IjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLyIzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
原方程變為:,即;
![齊次微分方程](/img/c/d93/wZwpmL2AzNycDM0IDN2EzM1UTM1QDN5MjM5ADMwAjMwUzLyQzL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
分離變數可得:,
![齊次微分方程](/img/4/2e5/wZwpmL2QjN5gDM1QTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL4UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
左右兩端同時積分可得:,
![齊次微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次微分方程](/img/a/6e7/wZwpmLyQTO2QDMwQDN2EzM1UTM1QDN5MjM5ADMwAjMwUzL0QzL3czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
將 代入,便可得到原方程的通解為:,其中 C 為任意常數。