一道微分方程解的问题设线性无关函数y1,y2,y3都是二阶非齐次线性方程y''+P(x)y'+Q(x)y=f(x)的解,C1 C2是待定常数.则此方程的通解是:A.C1y1+C2y2+y3 B.C1y1+C2y2-(C1+C2)y3C.C1y1+C2y2-(1-C1-C2)y3 D.C1y1+C2y2+(1-C1-C2)y3
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![一道微分方程解的问题设线性无关函数y1,y2,y3都是二阶非齐次线性方程y''+P(x)y'+Q(x)y=f(x)的解,C1 C2是待定常数.则此方程的通解是:A.C1y1+C2y2+y3 B.C1y1+C2y2-(C1+C2)y3C.C1y1+C2y2-(1-C1-C2)y3 D.C1y1+C2y2+(1-C1-C2)y3](/uploads/image/z/11953619-35-9.jpg?t=%E4%B8%80%E9%81%93%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E8%A7%A3%E7%9A%84%E9%97%AE%E9%A2%98%E8%AE%BE%E7%BA%BF%E6%80%A7%E6%97%A0%E5%85%B3%E5%87%BD%E6%95%B0y1%2Cy2%2Cy3%E9%83%BD%E6%98%AF%E4%BA%8C%E9%98%B6%E9%9D%9E%E9%BD%90%E6%AC%A1%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8By%27%27%2BP%28x%29y%27%2BQ%28x%29y%3Df%28x%29%E7%9A%84%E8%A7%A3%2CC1+C2%E6%98%AF%E5%BE%85%E5%AE%9A%E5%B8%B8%E6%95%B0.%E5%88%99%E6%AD%A4%E6%96%B9%E7%A8%8B%E7%9A%84%E9%80%9A%E8%A7%A3%E6%98%AF%3AA.C1y1%2BC2y2%2By3+B.C1y1%2BC2y2-%28C1%2BC2%29y3C.C1y1%2BC2y2-%281-C1-C2%29y3+D.C1y1%2BC2y2%2B%281-C1-C2%29y3)
一道微分方程解的问题设线性无关函数y1,y2,y3都是二阶非齐次线性方程y''+P(x)y'+Q(x)y=f(x)的解,C1 C2是待定常数.则此方程的通解是:A.C1y1+C2y2+y3 B.C1y1+C2y2-(C1+C2)y3C.C1y1+C2y2-(1-C1-C2)y3 D.C1y1+C2y2+(1-C1-C2)y3
一道微分方程解的问题
设线性无关函数y1,y2,y3都是二阶非齐次线性方程y''+P(x)y'+Q(x)y=f(x)的解,C1 C2是待定常数.则此方程的通解是:
A.C1y1+C2y2+y3 B.C1y1+C2y2-(C1+C2)y3
C.C1y1+C2y2-(1-C1-C2)y3 D.C1y1+C2y2+(1-C1-C2)y3
注:解释说y1-y3,y2-y3为对应齐次方程的特解,凭什么这么说呢,线性无关在这里又该怎么分解,
一道微分方程解的问题设线性无关函数y1,y2,y3都是二阶非齐次线性方程y''+P(x)y'+Q(x)y=f(x)的解,C1 C2是待定常数.则此方程的通解是:A.C1y1+C2y2+y3 B.C1y1+C2y2-(C1+C2)y3C.C1y1+C2y2-(1-C1-C2)y3 D.C1y1+C2y2+(1-C1-C2)y3
y1''+Py1'+Qy1=f
y3''+Py3'+Qy3=f
两式一减就得出y1-y3满足齐次方程y''+Py'+Qy=0,所以是齐次方程的特解.y2-y3的情况一样.
因为线性无关,所以y1-y3和y2-y3就构成了非齐次方程的基本解矩阵Φ,再加上一个特解y3,CΦ+y3就是非齐次方程的通解.
答案D.
答案是D
因为y1-y3,y2-y3都是y''+P(x)y'+Q(x)y=0的特解啊,
又y''+P(x)y'+Q(x)y=f(x)的通解就是y''+P(x)y'+Q(x)y=0的通解加上y''+P(x)y'+Q(x)y=f(x)的一个特解,C1(y1-y3)+C2(y2-y3)即为y''+P(x)y'+Q(x)y=0的通解,加上y1,y2,y3中任意一个即为y''+P(x)y'+Q(x)y=f(x)的通解