高等数学第十二章答案 同济五版121.docx

习题12?1

1?试说出下列各微分方程的阶数?

(1)x(y?)2?2yy??x?0?解一阶?(2)x2y??xy??y?0?解一阶?(3)xy????2y??x2y?0?解三阶?

(4)(7x?6y)dx?(x?y)dy?0?

解一阶?

(5)Ld2Q?RdQ?Q?0?

dt2 dt C

解二阶?

(6)d????sin2??

d?

解一阶?

2?指出下列各题中的函数是否为所给微分方程的解?

(1)xy??2y?y?5x2?

解y??10x?

因为xy??10x2?2(5x2)?2y?所以y?5x2是所给微分方程的解?

y??y?0?y?3sinx?4cosx?

解y??3cosx?4sinx?

因为y??y?3cosx?4sinx?3sinx?4cosx?7sinx?cosx?0?

所以y?3sinx?4cosx不是所给微分方程的解?

(3)y???2y??y?0?y?x2ex?

解y??2xex?x2ex?y???2ex?2xex?2xex?x2ex?2ex?4xex?x2ex?

因为y???2y??y?2ex?4xex?x2ex?2(2xex?x2ex)?x2ex?2ex?0?所以y?x2ex不是所给微分方程的解?

(4)y???(?

??)y????

y?0? y?Ce?x?C

e?x?

1 2 12

1 1 2 2

解y??C?e?x?C

?e?x? y??C?2e?x?C

?2e?x?

11 1

22 2

11 1

22 2

因为y??(???)y????y

1 2 12

?C?2e?x?C

?2e?x?(???

)(C?e?x?C

?e?x)???

(Ce?x?C

e?x)

11 1

?0?

22 2

1 2 11 1

22 2

12 1 1 2 2

所以y?Ce?x?Ce?x是所给微分方程的解?

1 1 2 2

3?在下列各题中?验证所给二元方程所确定的函数为所给微分方程的解?

(1)(x?2y)y??2x?y?x2?xy?y2?C?

解将x2?xy?y2?C的两边对x求导得

2x?y?xy??2yy??0?

即 (x?2y)y??2x?y?

所以由x2?xy?y2?C所确定的函数是所给微分方程的解?

(2)(xy?x)y???xy?2?yy??2y??0?y?ln(xy)?

解将y?ln(xy)的两边对x求导得

y??1?1y??即y????y ?

x y xy?x

再次求导得

y?(xy?x)?y(y?xy??1) ?xy??y2?y 1 x

y?? ? ?

(xy?x)2 (xy?x)2

注意到由y??1?1y?可得xy??xy??1?所以

x y y

?(?

xy?x y

y?2?yy??y?)?

y??? 1 ?[?(xy??1)y??yy??y?]? 1 ?(?xy?2?yy??2y?)?

xy?x xy?x

从而 (xy?x)y???xy?2?yy??2y??0?

即由y?ln(xy)所确定的函数是所给微分方程的解?

4?在下列各题中?确定函数关系式中所含的参数?使函数满足所给的初始条件?

(1)x2?y2?C?y|

x

?5?

?0y

?0

y|

x?0

?0得02?52?C?C??25?故x2?y2??25?

(2)y?(C1?C2x)e2x?y|??0?y?|

?1?

x0 x?0

2 1 2解y??Ce2x?2(C?Cx)e2x

2 1 2

y|

y|

x?0

?0?y?|

?0

?0

?1得

?1 ??C

?1 ?

?C?C?1

?

2 1

1 2解之得C?0?C?1?故y?xe2x

1 2

1 2y?Csin(x?C)?y

1 2

x

???1?y?|

??0?

x?1 2解y??Ccos(x?C

x?

1 2

由y|

x

???1?y?|

??0得

x??Csin?(?

x?

)?1

?CsinC?1

?1 ? 2

?即?1 2 ?

?Ccos(?C)?0 ??CcosC?0

1 2 1 2

解之得C

?1? C???故y?sin(x??)?