HMMT 二月 2010 · CALC 赛 · 第 9 题

英文解析

9. [ 7 ] Let x ( t ) be a solution to the differential equation
   ′ ′′2
   ( x + x ) + x · x = cos t
   √
   ( )
   2 π
   ′
   with x (0) = x (0) = . Compute x .
   5 4
   √
   4504
   2 ′ ′ ′ 2 ′ ′
   Answer: Rewrite the equation as x + 2 xx + ( xx ) = cos t . Let y = x , so y = 2 xx and
   ′ 1 ′′5
   the equation becomes y + y + y = cos t . The term cos t suggests that the particular solution should
   4 22
   be in the form A sin t + B cos t . By substitution and coefficient comparison, we get A = and B = .
   5 5
   4 2 2
   Since the function y ( t ) = sin t + cos t already satisfies the initial conditions y (0) = x (0) = and 2
   5 5 5
   ′ ′4
   y (0) = 2 x (0) x (0) = , the function y also solves the initial value problem. Note that since x is
   ( )5
   π πpositive at t = 0 and y = x never reaches zero before t reaches , the value of x must be positive.2
   4 4
   √
   √
   √ √
   ( ) ( )
   π π 6 2 4504
   Therefore, x = + y = · = .
   4 4 5 2 5
   ∑n 1