基本介紹
![齊次一階微分方程](/img/3/895/wZwpmLyAzNwUTO5QTMwEDN0UTMyITNykTO0EDMwAjMwUzL0EzL2gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/4/90d/wZwpmL1UTN0ATM3kzNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5czLwczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/9/806/wZwpmL1cDMxgDN2UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLxAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/a/342/wZwpmLwUDO0IzMygjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4YzLzYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/5/149/wZwpmL3MTM4MTNxEDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLxQzL0YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/9/806/wZwpmL1cDMxgDN2UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLxAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/2/e5e/wZwpmL0QTNwAjMzQzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czLwYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
如果對任何都有,則稱是x和y的 齊次函式,如果取,則。這就是說齊次函式可改寫為的形式。
![齊次一階微分方程](/img/4/5e0/wZwpmLwADOwYjMzMjNwMzM1UTM1QDN5MjM5ADMwAjMwUzLzYzLyEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/9/806/wZwpmL1cDMxgDN2UzMzEzM1UTM1QDN5MjM5ADMwAjMwUzL1MzLxAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
一階微分方程(其中,為齊次函式)就叫做 齊次(一階微分)方程。或者說,方程
![齊次一階微分方程](/img/d/df8/wZwpmL4ATN5MjN3QjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzL0EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
是齊次方程。此外,如果在微分方程的每一項中,因子x和y的冪次的總和都是相等的,則該方程就是 齊次方程。
例如
![齊次一階微分方程](/img/7/843/wZwpmL3QjN5gDNyUjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL1YzL4MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/5/1c2/wZwpmL4gDN3ITOwUDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1gzLygzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/e/099/wZwpmL0UDOyEDMxMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzL0czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
都是齊次方程。事實上,式(2)各項同除x,式(3)各項同除以,則式(2)和(3)可分別化為
![齊次一階微分方程](/img/0/fbe/wZwpmL3cDO2IDM3cTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3UzLyAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
或
![齊次一階微分方程](/img/b/0f5/wZwpmLzUDM5QzN5QjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0YzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
和
![齊次一階微分方程](/img/d/9de/wZwpmL4cDN3MTN2MDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLzQzL1gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
或
![齊次一階微分方程](/img/8/add/wZwpmL1gDOwETN2QDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0QzLxEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
另外,方程
![齊次一階微分方程](/img/8/356/wZwpmL3gjN3IzM2QTN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL4YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
也是齊次方程。事實上,方程(4)右端分子和分母同除以x,則得到齊次方程
![齊次一階微分方程](/img/2/e22/wZwpmLyMTMxcDMxUDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1gzLwYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
一般解法
![齊次一階微分方程](/img/7/038/wZwpmLzUjM5YzNwgjN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4YzLxUzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
關於齊次方程的一般解法如下 :
![齊次一階微分方程](/img/b/508/wZwpmLwETN4MDO4ADN3UzM1UTM1QDN5MjM5ADMwAjMwUzLwQzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/2/d1d/wZwpmLxQDOzkDOzIDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzL2gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
令所以,代入方程(1),得
![齊次一階微分方程](/img/7/e90/wZwpmL3ATN2QTO2QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czL1AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
即有
![齊次一階微分方程](/img/8/306/wZwpmL4gTMyQDOyMDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzAzL4MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
方程(2)為可分離變數方程,於是
![齊次一階微分方程](/img/2/c41/wZwpmLwQjMwYjM0UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzL3QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
方程(3)兩端積分,得
![齊次一階微分方程](/img/d/926/wZwpmL4UzM1ATN2gTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL4UzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
上述等式可改寫為
![齊次一階微分方程](/img/7/15c/wZwpmLyATO1ADNwQDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
把代入式(4),則得到方程(1)的隱式通解
![齊次一階微分方程](/img/8/3d2/wZwpmLzADN4czMwEjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLxYzL0MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/e/ff7/wZwpmLwgzNzIjM0ADN3UzM1UTM1QDN5MjM5ADMwAjMwUzLwQzLwQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
例1求方程的通解。
![齊次一階微分方程](/img/e/cc9/wZwpmLyEzN5gDN4UDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL1QzLwczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/6/02b/wZwpmLxYzN4MDM3AjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLwIzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/5/417/wZwpmL1EzN2ADOxQDN3UzM1UTM1QDN5MjM5ADMwAjMwUzL0QzL0YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/b/e11/wZwpmL1cjN4cjN4YzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL3QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/f/e34/wZwpmLxEjM2UjM3QDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0gzLyQzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/7/571/wZwpmLwITN5ATNyQDM3UzM1UTM1QDN5MjM5ADMwAjMwUzL0AzL0QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/1/c35/wZwpmL2UzMxgzN3UzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL1MzL3EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/8/620/wZwpmLxcTN3UzMykTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL2MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/4/583/wZwpmLzETM1gjN4UTO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1kzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/e/444/wZwpmL3cDMzgjMwYTN2UzM1UTM1QDN5MjM5ADMwAjMwUzL2UzL1IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![齊次一階微分方程](/img/6/ab2/wZwpmL1EzNyMDO2IjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLyYzL3IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
解:方程,令,所以,於是方程變為,即,所以。積分得通解,即。也可以把方程的隱式通解改寫為顯式通解。事實上,因為,所以。