题目来自浙江大学出版社蔡燧林编第二版常微分方程设p(x)q(x)f(x)为连续函数,且f(x)不等于0,y1(x)y2(x)y3(x)为y''+p(x)y'+q(x)y=f(x)的三个解.(E)对应的齐次方程记为(E0).并设abc为3个常数,y(x)=ay1(x)+by2(x)
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![题目来自浙江大学出版社蔡燧林编第二版常微分方程设p(x)q(x)f(x)为连续函数,且f(x)不等于0,y1(x)y2(x)y3(x)为y''+p(x)y'+q(x)y=f(x)的三个解.(E)对应的齐次方程记为(E0).并设abc为3个常数,y(x)=ay1(x)+by2(x)](/uploads/image/z/13317044-68-4.jpg?t=%E9%A2%98%E7%9B%AE%E6%9D%A5%E8%87%AA%E6%B5%99%E6%B1%9F%E5%A4%A7%E5%AD%A6%E5%87%BA%E7%89%88%E7%A4%BE%E8%94%A1%E7%87%A7%E6%9E%97%E7%BC%96%E7%AC%AC%E4%BA%8C%E7%89%88%E5%B8%B8%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E8%AE%BEp%28x%29q%28x%29f%28x%29%E4%B8%BA%E8%BF%9E%E7%BB%AD%E5%87%BD%E6%95%B0%2C%E4%B8%94f%28x%29%E4%B8%8D%E7%AD%89%E4%BA%8E0%2Cy1%28x%29y2%28x%29y3%28x%29%E4%B8%BAy%27%27%2Bp%28x%29y%27%2Bq%28x%29y%3Df%28x%29%E7%9A%84%E4%B8%89%E4%B8%AA%E8%A7%A3%EF%BC%8E%28E%29%E5%AF%B9%E5%BA%94%E7%9A%84%E9%BD%90%E6%AC%A1%E6%96%B9%E7%A8%8B%E8%AE%B0%E4%B8%BA%28E0%29%EF%BC%8E%E5%B9%B6%E8%AE%BEabc%E4%B8%BA3%E4%B8%AA%E5%B8%B8%E6%95%B0%2Cy%28x%29%3Day1%28x%29%2Bby2%28x%29)
题目来自浙江大学出版社蔡燧林编第二版常微分方程设p(x)q(x)f(x)为连续函数,且f(x)不等于0,y1(x)y2(x)y3(x)为y''+p(x)y'+q(x)y=f(x)的三个解.(E)对应的齐次方程记为(E0).并设abc为3个常数,y(x)=ay1(x)+by2(x)
题目来自浙江大学出版社蔡燧林编第二版常微分方程
设p(x)q(x)f(x)为连续函数,且f(x)不等于0,y1(x)y2(x)y3(x)为y''+p(x)y'+q(x)y=f(x)的三个解.(E)对应的齐次方程记为(E0).并设abc为3个常数,y(x)=ay1(x)+by2(x)+cy3(x).(1)证明y(x)为(E)的解的充要条件是a+b+c=1;y(x)为(E0)的解的充要条件是a+b+c=0.(2)若增设y1(x)y2(x)y3(X)线性无关,abc中两个为任意常数,证明y(x)为(E)的解的充要条件是a+b+c=1;y(x)为(E0)的解的充要条件是a+b+c=0.
第二小题为(2)若增设y1(x)y2(x)y3(X)线性无关,abc中两个为任意常数,证明y(x)为(E)的 的充要条件是a+b+c=1;y(x)为(E0)的 的充要条件是a+b+c=0
题目来自浙江大学出版社蔡燧林编第二版常微分方程设p(x)q(x)f(x)为连续函数,且f(x)不等于0,y1(x)y2(x)y3(x)为y''+p(x)y'+q(x)y=f(x)的三个解.(E)对应的齐次方程记为(E0).并设abc为3个常数,y(x)=ay1(x)+by2(x)
第一题直接代进去
y''+p(x)y'+q(x)y = (a+b+c)f(x)
然后就显然了
第二题注意到y1(x)-y2(x)和y1(x)-y3(x)是(E0)的两个线性无关解,通解可由它们线性组合得到