권찡's 공학이야기

미분방정식(ODE) 풀이-프로베니우스 해법 본문

mathematics/engineering math

미분방정식(ODE) 풀이-프로베니우스 해법

권찡 2018. 11. 8. 17:27

프로베니우스 해법은 앞서 다룬 급수해법중에서 어떠한 경우를 나타냅니다.


물리학이나 공학에서 많이 나타나는 미분방정식은 아래의 형태를 많이 가집니다.


a(x)y''%2Bb(x)y'%2Bc(x)y%3D0%20



이때 a(x)를 양변에 나누며 정리하면 아래의 형태가 됩니다.


%5C%5C%20y''%2Bp(x)y'%2Bq(x)y%3D0%20




이때 각각 p,q의 분모 가 0이 되지 않는다면 정상해를 갖고, 0이 된다면 특이해를 갖는다고 정리했습니다.


세부적으로 들어가면 정칙 특이점, 비정칙 특이점이 존재합니다.



예로

%EC%BD%94%EC%8B%9C%5Cquad%20%EC%98%A4%EC%9D%BC%EB%9F%AC%5Cquad%20%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%84%5Cquad%20%EC%98%88%EB%A1%9C%5Cquad%20%EB%93%A4%EA%B2%A0%EC%8A%B5%EB%8B%88%EB%8B%A4.%5C%5C%20%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7Dy''%2Baxy'%2Bby%3D0%5Cquad%20%EC%97%90%EC%84%9C%5Cquad%20x%3D0%EC%97%90%EC%84%9C%5Cquad%20%EC%A0%95%EC%B9%99%5Cquad%20%ED%8A%B9%EC%9D%B4%EC%A0%90%EC%9D%84%5Cquad%20%EA%B0%96%EB%8A%94%EB%8B%A4%5C%5C%20%5C%5C%20%EC%9C%84%5Cquad%20%EC%82%AC%EC%8B%A4%EC%9D%84%5Cquad%20%EC%A6%9D%EB%AA%85%ED%95%98%EA%B2%A0%EC%8A%B5%EB%8B%88%EB%8B%A4.%5Cquad%20%EC%96%91%5Cquad%20%EB%B3%80%EC%9D%84%5Cquad%20%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D%EC%9C%BC%EB%A1%9C%5Cquad%20%EB%82%98%EB%88%84%EB%A9%B4%5C%5C%20%5C%5C%20y''%2B%5Cfrac%20%7B%20a%20%7D%7B%20x%20%7Dy'%2B%5Cfrac%20%7B%20b%20%7D%7B%20%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D%20%7Dy%3D0%5Cquad%20%EC%9D%B4%5Cquad%20%EB%90%A9%EB%8B%88%EB%8B%A4.%5Cquad%20%EC%9D%B4%EB%95%8C%5Cquad%20p%2Cq%EC%9E%90%EB%A6%AC%EC%97%90%5Cquad%20%EA%B0%81%EA%B0%81%5Cquad%20x%EA%B0%80%5Cquad%200%EC%9D%B4%5Cquad%20%EB%90%98%EA%B2%8C%ED%95%98%EB%8A%94%5Cquad%20%EC%9D%B8%EC%9E%90%EA%B0%80%5C%5C%20%5C%5C%20%EA%B8%B0%EA%BB%8F%ED%95%B4%EC%95%BC%5Cquad%201%EC%B0%A8%2C%5Cquad%20%EA%B7%B8%EB%A6%AC%EA%B3%A0%5Cquad%202%EC%B0%A8%5Cquad%20%EC%9D%B8%EA%B2%BD%EC%9A%B0%5Cquad%20x%3D0%EC%97%90%EC%84%9C%5Cquad%20%EC%A0%95%EC%B9%99%5Cquad%20%ED%8A%B9%EC%9D%B4%EC%A0%90%EC%9D%84%5Cquad%20%EA%B0%80%EC%A7%80%EB%8A%94%5Cquad%20%EA%B2%83%EC%9E%85%EB%8B%88%EB%8B%A4.%5C%5C%20%5C%5C%20%5C%5C%20%EB%B0%98%EB%8C%80%EB%A1%9C%5Cquad%20%EB%B9%84%EC%A0%95%EC%B9%99%5Cquad%20%ED%8A%B9%EC%9D%B4%EC%A0%90%EC%9D%80%5C%5C%20(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-4)y''%2B(x-2)y'%2By%3D0%5Cquad%20%EC%97%90%EC%84%9C%5Cquad%20x%3D2%EC%97%90%EC%84%9C%5Cquad%20%EC%A0%95%EC%B9%99%5Cquad%20x%3D-2%EC%97%90%EC%84%9C%5Cquad%20%EB%B9%84%EC%A0%95%EC%B9%99%EC%9D%84%5Cquad%20%EA%B0%80%EC%A7%91%EB%8B%88%EB%8B%A4.%5C%5C%20%5C%5C%20a(x)%EC%9D%B8%5Cquad%20(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-4)%EB%A5%BC%5Cquad%20%EB%82%98%EB%88%A0%EB%B3%B4%EB%A9%B4%5C%5C%20y''%2B%5Cfrac%20%7B%20x-2%20%7D%7B%20(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-4)%20%7Dy'%2B%5Cfrac%20%7B%201%20%7D%7B%20(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-4)%20%7Dy%3D0%5C%5C%20%5C%5C%20%EC%97%AC%EA%B8%B0%EC%84%9C%5Cquad%20%EB%B3%B4%EB%A9%B4%5Cquad%20x-2%EB%8A%94%5Cquad%20%EB%B6%84%EB%AA%A8%EC%9E%90%EB%A6%AC%EC%97%90%5Cquad%20%ED%95%98%EB%82%98%EC%94%A9%EB%A7%8C%5Cquad%20%EC%9E%88%EC%96%B4%EC%84%9C%5Cquad%20%EC%A0%95%EC%B9%99%EC%9D%B4%EA%B3%A0%5Cquad%20x%2B2%EB%8A%94%5Cquad%20%EC%96%91%EB%B3%80%EC%97%90%5Cquad%202%EC%B0%A8%EC%94%A9%5Cquad%20%EC%9E%88%EC%96%B4%EC%84%9C%5C%5C%20%EB%B9%84%EC%A0%95%EC%B9%99%EC%9E%85%EB%8B%88%EB%8B%A4.%20




이전에 급수해법을 설명할때 특이해를 가질경우 어떻게 급수를 정하는가에 대해서 잠깐 보여드렸습니다.



y%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20C%20%7D%5Ccombi%20%5E%7B%20n%2Br%20%7D%7B%20x%20%7D%20%7D%20



문제를 가지고 이해해봅시다.



%5C%5C%20%EC%98%88%EB%A5%BC%5Cquad%20%EB%93%A4%EC%96%B4%EC%84%9C%5Cquad%20%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7Dy''%2Bxy'%2B(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-%5Ccombi%20%5E%7B%202%20%7D%7B%20v%20%7D)y%3D0%5Cquad%20%EC%9D%B4%EB%9F%AC%ED%95%9C%5Cquad%20%EB%AF%B8%EB%B0%A9%EC%9D%B4%5Cquad%20%EC%A1%B4%EC%9E%AC%5Cquad%20%ED%95%A0%EB%95%8C%5Cquad%20%5C%5C%20%5C%5C%20%EC%9C%84%EC%9D%98%5Cquad%20%EC%A0%95%EB%A6%AC%EC%97%90%5Cquad%20%EB%94%B0%EB%A5%B4%EB%A9%B4%5Cquad%20%ED%8A%B9%EC%9D%B4%EC%A0%90%5Cquad%20%EC%A6%89%5Cquad%20%ED%8A%B9%EC%9D%B4%ED%95%B4%EB%A5%BC%5Cquad%20%EA%B0%96%EB%8A%94%EB%8B%A4%EB%8A%94%5Cquad%20%EA%B2%83%EC%9D%84%5Cquad%20%EC%95%8C%EC%88%98%5Cquad%20%EC%9E%88%EC%8A%B5%EB%8B%88%EB%8B%A4.%5C%5C%20y%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D%5Ccombi%20%5E%7B%20n%2Br%20%7D%7B%20x%20%7D%20%7D%EC%9D%B4%5Cquad%20%ED%8A%B9%EC%9D%B4%ED%95%B4%EB%A5%BC%5Cquad%20%EA%B0%80%EC%A7%88%EB%95%8C%5Cquad%20%EA%B8%89%EC%88%98%ED%95%B4%EC%9D%98%5Cquad%20%ED%98%95%ED%83%9C%EC%9D%B4%EA%B8%B0%EB%95%8C%EB%AC%B8%EC%97%90%5Cquad%20%EC%9C%84%5Cquad%20%EC%8B%9D%EC%97%90%5Cquad%20%EB%8C%80%EC%9E%85%ED%95%B4%5Cquad%20%EB%B4%85%EC%8B%9C%EB%8B%A4%5C%5C%20y'%3D%5Csum%20_%7B%20n%3D1%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D(n%2Br)%5Ccombi%20%5E%7B%20n%2Br-1%20%7D%7B%20x%20%7D%20%7D%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%5Cquad%20%2C%5Cquad%20%5Cquad%20%5Cquad%20y''%3D%5Csum%20_%7B%20n%3D2%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D(n%2Br)(n%2Br-1)%5Ccombi%20%5E%7B%20n%2Br-2%20%7D%7B%20x%20%7D%20%7D%5C%5C%20%EB%94%B0%EB%9D%BC%EC%84%9C%5C%5C%20%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D(n%2Br)(n%2Br-1)%5Ccombi%20%5E%7B%20n%2Br%20%7D%7B%20x%20%7D%20%7D%2B%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D(n%2Br)%5Ccombi%20%5E%7B%20n%2Br%20%7D%7B%20x%20%7D%20%7D%2B(%5Ccombi%20%5E%7B%202%20%7D%7B%20x%20%7D-%5Ccombi%20%5E%7B%202%20%7D%7B%20v%20%7D)%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D%5Ccombi%20%5E%7B%20n%2Br%20%7D%7B%20x%20%7D%20%7D%3D0%5C%5C%20%5C%5C%20%EC%9D%B4%EB%A0%87%EA%B2%8C%5Cquad%20%EC%A0%95%EB%A6%AC%5Cquad%20%ED%95%A0%EC%88%98%5Cquad%20%EC%9E%88%EC%8A%B5%EB%8B%88%EB%8B%A4.%20




%5C%5C%20%EC%9D%B4%EB%95%8C%5Cquad%20n%3D0%5Cquad%20%EB%8C%80%EC%9E%85%ED%96%88%EC%9D%84%EB%95%8C%5Cquad%20%5Ccombi%20%5E%7B%20r%20%7D%7B%20x%20%7D%EC%9D%98%5Cquad%20%EA%B3%84%EC%88%98%EB%A7%8C%5Cquad%20%EB%B4%85%EC%8B%9C%EB%8B%A4.%5C%5C%20%EA%B3%84%EC%82%B0%ED%95%B4%EB%B3%B4%EB%A9%B4%5Cquad%20(%5Ccombi%20%5E%7B%202%20%7D%7B%20r%20%7D-%5Ccombi%20%5E%7B%202%20%7D%7B%20v%20%7D)%EC%9D%B4%5Cquad%20%EA%B3%84%EC%88%98%EB%9D%BC%EB%8A%94%5Cquad%20%EA%B2%83%EC%9D%84%5Cquad%20%EC%95%8C%EC%88%98%5Cquad%20%EC%9E%88%EC%8A%B5%EB%8B%88%EB%8B%A4.%5C%5C%20%5C%5C%20%EC%9D%B4%5Cquad%20%EC%95%9E%EC%9D%98%5Cquad%20%EA%B3%84%EC%88%98%EA%B0%80%5Cquad%20r%EC%97%90%EB%8C%80%ED%95%9C%5Cquad%20%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%B4%EB%9D%BC%EA%B3%A0%5Cquad%20%EB%B3%B4%EB%A9%B4%5Cquad%20%EC%9D%B4%EC%8B%9D%EC%9D%B4%5Cquad%20%EB%B0%94%EB%A1%9C%5Cquad%20%EA%B2%B0%EC%A0%95%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9E%85%EB%8B%88%EB%8B%A4.%5C%5C%20%5C%5C%20%EC%9D%B4%5Cquad%20%EA%B2%B0%EC%A0%95%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%B4%5Cquad%20%EB%AC%B4%EC%8A%A8%5Cquad%20%EC%9D%98%EB%AF%B8%EB%A5%BC%5Cquad%20%EA%B0%80%EC%A7%80%EB%83%90%EB%A9%B4%5Cquad%20%EC%9D%B4%5Cquad%20%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%84%5Cquad%20%ED%92%80%EC%96%B4%EC%84%9C%5Cquad%20r%EC%9D%84%5Cquad%20%EA%B2%B0%EC%A0%95%ED%95%98%EA%B3%A0%5Cquad%20%EC%88%9C%ED%99%98%EA%B4%80%EA%B3%84%EB%A5%BC%5Cquad%20%5C%5C%20%EC%82%B4%ED%8E%B4%EB%B3%B4%EB%A9%B4%5Cquad%20%ED%95%B4%EB%A5%BC%5Cquad%20%EC%95%8C%EC%88%98%5Cquad%20%EC%9E%88%EC%8A%B5%EB%8B%88%EB%8B%A4.%5Cquad%20%EB%B3%B4%ED%86%B5%5Cquad%20%EA%B2%B0%EC%A0%95%5Cquad%20%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%80%5Cquad%20r%EC%97%90%5Cquad%20%EB%8C%80%ED%95%9C%5Cquad%202%EC%B0%A8%EC%8B%9D%EC%9D%B8%EB%8D%B0%5Cquad%20%EA%B7%BC%EC%9D%98%5Cquad%20%ED%98%95%ED%83%9C%EC%97%90%5Cquad%20%EB%94%B0%EB%9D%BC%EC%84%9C%5Cquad%20%5C%5C%20%EB%AF%B8%EB%B0%A9%EC%9D%98%5Cquad%20%ED%95%B4%EC%9D%98%5Cquad%20%ED%98%95%ED%83%9C%EB%A5%BC%5Cquad%20%EA%B2%B0%EC%A0%95%ED%95%A0%EC%88%98%5Cquad%20%EC%9E%88%EC%8A%B5%EB%8B%88%EB%8B%A4.%5Cquad%20%EC%A6%9D%EB%AA%85%EC%9D%80%5Cquad%20%EC%83%9D%EB%9E%B5%ED%95%98%EA%B2%A0%EC%8A%B5%EB%8B%88%EB%8B%A4.%20






%EB%A7%8C%EC%95%BD%5Cquad%20%EA%B2%B0%EC%A0%95%5Cquad%20%EB%B0%A9%EC%A0%95%EC%8B%9D%EC%9D%98%5Cquad%20%EA%B7%BC%EC%9D%B4%5Cquad%20a%2Cb%EB%9D%BC%EB%A9%B4%5C%5C%201%5D%5Cquad%20a-b%EA%B0%80%5Cquad%20%EC%A0%95%EC%88%98%EA%B0%80%5Cquad%20%EC%95%84%EB%8B%8C%5Cquad%20%EA%B2%BD%EC%9A%B0%EB%8A%94%5Cquad%20%EC%9D%BC%EC%B0%A8%5Cquad%20%EB%8F%85%EB%A6%BD%EC%9D%98%5Cquad%20%ED%95%B4%EB%8B%A4%5C%5C%20%EC%A6%89%5Cquad%20%5Ccombi%20_%7B%201%20%7D%7B%20y%20%7D%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D%5Ccombi%20%5E%7B%20n%2Ba%20%7D%7B%20x%20%7D%20%7D%5Cquad%20%2C%5Cquad%20%5Ccombi%20_%7B%202%20%7D%7B%20y%20%7D%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20d%20%7D%5Ccombi%20%5E%7B%20n%2Bb%20%7D%7B%20x%20%7D%20%7D%5C%5C%202%5Da-b%EA%B0%80%5Cquad%20%EC%96%91%EC%9D%98%5Cquad%20%EC%A0%95%EC%88%98%EC%9D%BC%5Cquad%20%EA%B2%BD%EC%9A%B0%5Cquad%20%EC%97%AD%EC%8B%9C%5Cquad%20%EB%8F%85%EB%A6%BD%EC%9D%B4%EA%B3%A0%5Cquad%20k%EA%B0%80%5Cquad%200%EC%9D%BC%EC%88%98%5Cquad%20%EC%9E%88%EB%8B%A4.%5C%5C%20%5Ccombi%20_%7B%201%20%7D%7B%20y%20%7D%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D%5Ccombi%20%5E%7B%20n%2Ba%20%7D%7B%20x%20%7D%20%7D%5Cquad%20%2C%5Cquad%20%5Cquad%20%5Ccombi%20_%7B%202%20%7D%7B%20y%20%7D%3Dk%5Ccombi%20_%7B%201%20%7D%7B%20y%20%7Dlnx%2B%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20d%20%7D%5Ccombi%20%5E%7B%20n%2Bb%20%7D%7B%20x%20%7D%20%7D%5C%5C%203%5Da%3Db%5Cquad%20%EA%B2%BD%EC%9A%B0%5Cquad%20%5C%5C%20%5Ccombi%20_%7B%201%20%7D%7B%20y%20%7D%3D%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20c%20%7D%5Ccombi%20%5E%7B%20n%2Ba%20%7D%7B%20x%20%7D%20%7D%5Cquad%20%2C%5Cquad%20%5Cquad%20%5Ccombi%20_%7B%202%20%7D%7B%20y%20%7D%3D%5Ccombi%20_%7B%201%20%7D%7B%20y%20%7Dlnx%2B%5Csum%20_%7B%20n%3D0%20%7D%5E%7B%20%5Cinfty%20%20%7D%7B%20%5Ccombi%20_%7B%20n%20%7D%7B%20d%20%7D%5Ccombi%20%5E%7B%20n%2Bb%20%7D%7B%20x%20%7D%20%7D%20




이러한 방식으로 특이해를 구할수 있습니다.





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