问题 问答题

设函数f(x)满足f″(x)+2f′(x)-3f(x)=2ex,求微分方程的一个特解函数f(x).

答案

参考答案:

[考点点击] 主要考查的知识点为二阶常系数线性非齐次微分方程.

[要点透析] 此方程是二阶常系数非齐次微分方程,其中f(x)=2ex属于eλxPm(x)型λ=1,m=0),则该方程所对应的齐次方程的特征方程为r2+2r-3=0,解特征根为r1=-3,r2=1,所以λ=1是对应齐次方程的特征根,且为单根.

故设其特解为f(x)*=b0xex

则f(x)*′=b0ex+b0xex

f(x)*″=b0ex+b0ex+b0xex=2b0ex+b0xex

代入微分方程得

于是原微分方程的一个特解

单项选择题
阅读理解

阅读理解。

     Every day, putting food on the table is a challenge for many Americans. They worry that they won't be

able to feed their families. The slowing economy, increasing unemployment and rising food prices are

contributing to the growing number of people who need help. The nation's food banks are rising to the

challenge. It is estimated that they are currently serving 38% more people than they were a year ago.

     "In Minnesota, the jobless rate is the highest it's been in many years," says Tricia Theurer of Second

Harvest Heartland, a hunger-relief organization in Minnesota."More and more people are needing to ask for

help for emergency food, many for the first time, due to economic situations."

     Second Harvest Heartland provides about 41.5 million pounds of food annually to 170,000 people in 59

counties in Minnesota and Western Wisconsin. The group is part of Feeding America, the nation's food-bank

network. Its members include 205 food banks serving every county in the United States.

     Hunger affects children, adults and seniors. It affects those who live in the city, in the suburbs and in

rural areas. It affects those who do not have jobs and the working poor. The working poor are the fastest

growing group of food-bank clients. They are employed, but simply do not have enough income to buy

enough food to feed their families.

     Kids and adults can participate together in the fight to end hunger. The money you donate to the food bank

may seem like too small an amount to make a difference, but a simple $1 donation can be used to distribute

more than $9 worth of food. "Kids can be very creative in coming up with ideas of how to help others," says

Theurer."Some kids run lemonade stands, others have asked guests to contribute food or money to Second

Harvest instead of presents."

1. Why more and more people need to be helped in the USA?

A. Because putting food on the table is a challenge for many Americans.

B. Because of more people out of work and food prices rising with economy slowing.

C. Because more and more farmers give up planting in the economy crisis.

D. Because the clerks in nation's food banks were out of work at present.

2. What caused many people demand urgent food help for the first time?

A. The challenge of the nation's food banks.

B. The development of the internet.

C. The development of Second Harvest Heartland.

D. The worsening economic states.

3. What population are more in need in food-bank lists?

A. Seniors in cities.

B. People in rural areas.

C. The poor employees.

D. Children in schools.