3f(x)+xf'(x)=5x平方+2x-3,f(x)为多项式函数, 求f(x)

问题描述:

3f(x)+xf'(x)=5x平方+2x-3,f(x)为多项式函数, 求f(x)

设f(x) = ax² + bx + c
则 f'(x) = 2ax + b
3f(x) + xf'(x) = 5x² + 2x - 3
3ax² + 3bx + 3c + 2ax² + bx = 5x² + 2x - 3
5ax² + 4bx + 3c = 5x² + 2x - 3
所以 5a = 5,4b = 2,3c = -3
所以 a = 1,b = 1/2,c = -1
所以 f(x) = x² + x/2 - 1