设![](https://wkrtcs.bdimg.com/rtcs/image?w=309.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"309.33","h":"16.00","dataType":"gif","c":"word/media/image1.gif"})
Part1
常系数齐次微分方程![](https://wkrtcs.bdimg.com/rtcs/image?w=62.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"62.67","h":"16.00","dataType":"gif","c":"word/media/image2.gif"})
型方程通解:
Step1
求解特征值方程![](https://wkrtcs.bdimg.com/rtcs/image?w=289.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"289.33","h":"16.00","dataType":"gif","c":"word/media/image4.gif"})
，解得r值；
Step2
对比特征值写出每个特征值所对应的解集：
1 r是单实根（同一个实数对应1个r值）：
![](https://wkrtcs.bdimg.com/rtcs/image?w=74.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"74.67","h":"16.00","dataType":"gif","c":"word/media/image5.gif"})
（如![](https://wkrtcs.bdimg.com/rtcs/image?w=101.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"101.33","h":"16.00","dataType":"gif","c":"word/media/image6.gif"})
中r的解为，其中为特征方程的两个单实根。）
②r是K重实根（同一个实数对应k个r值）：
![](https://wkrtcs.bdimg.com/rtcs/image?w=197.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"197.33","h":"16.00","dataType":"gif","c":"word/media/image9.gif"})
（如![](https://wkrtcs.bdimg.com/rtcs/image?w=134.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"134.67","h":"16.00","dataType":"gif","c":"word/media/image10.gif"})
中r的解为，其中为特征方程的三个三重实根。）
③r是单复根（同一个复数对应1个r值），![](https://wkrtcs.bdimg.com/rtcs/image?w=73.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"73.33","h":"14.67","dataType":"gif","c":"word/media/image13.gif"})
:
![](https://wkrtcs.bdimg.com/rtcs/image?w=184.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"184.00","h":"17.33","dataType":"gif","c":"word/media/image14.gif"})
④r是k重复根，（同一个复数对应k个r值），:
![](https://wkrtcs.bdimg.com/rtcs/image?w=418.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"418.67","h":"16.00","dataType":"gif","c":"word/media/image15.gif"})
Step3
写出型方程通解
设
Part2
常系数非齐次微分方程![](https://wkrtcs.bdimg.com/rtcs/image?w=264.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"264.00","h":"36.00","dataType":"gif","c":"word/media/image17.gif"})
型方程通解
其中，![](https://wkrtcs.bdimg.com/rtcs/image?w=60.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"60.00","h":"36.00","dataType":"gif","c":"word/media/image18.gif"})
是x的m次多项式，包含m+1个常系数；是x的n次多项式，包含n+1个常系数。
Step1
观察方程的结构，得到的值，写出该方程的一个特解的结构：
![](https://wkrtcs.bdimg.com/rtcs/image?w=274.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_0","w":"274.67","h":"36.00","dataType":"gif","c":"word/media/image21.gif"})
Step2
确定特解的系数，其中，而![](https://wkrtcs.bdimg.com/rtcs/image?w=61.33&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"61.33","h":"36.00","dataType":"gif","c":"word/media/image22.gif"})
是x的d次多项式，包含d+1个常系数，，
![](https://wkrtcs.bdimg.com/rtcs/image?w=28.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"28.00","h":"16.00","dataType":"gif","c":"word/media/image24.gif"})
的取值如下：
1 如果![](https://wkrtcs.bdimg.com/rtcs/image?w=52.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"52.00","h":"17.33","dataType":"gif","c":"word/media/image25.gif"})
且分别为特征方程的：
1、 ![](https://wkrtcs.bdimg.com/rtcs/image?w=28.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"28.00","h":"16.00","dataType":"gif","c":"word/media/image26.gif"})
不是特征方程的实根，则k=0
2、 是特征方程的单实根，则k=1
3、 是特征方程的b实根，则k=b
2 如果![](https://wkrtcs.bdimg.com/rtcs/image?w=52.00&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"52.00","h":"17.33","dataType":"gif","c":"word/media/image28.gif"})
且分别为特征方程的：
1、![](https://wkrtcs.bdimg.com/rtcs/image?w=54.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_1","w":"54.67","h":"16.00","dataType":"gif","c":"word/media/image29.gif"})
不是特征方程的复根，则k=0
2、是特征方程的单复根，则k=1
3、是特征方程的b重复根，则k=b
Step3
按照Part1的步骤写出常系数非齐次微分方程对应的常系数齐次微分方程的通解。
Step4
常系数非齐次微分方程的通解为：
![](https://wkrtcs.bdimg.com/rtcs/image?w=94.67&md5sum=67d37028846b7b34e507c068afa75bda&sign=b0f2017bce&rtcs_flag=2&rtcs_ver=3.1&l=webapp&bucketNum=53&ipr={"t":"img","r":"r_3","w":"94.67","h":"16.00","dataType":"gif","c":"word/media/image30.gif"})
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