Question

常微分方程，求解方程组 x=2y-x+1 y=3y-2x

求老师解答

Answer 1

如图所示：

Answer 2

dx/dt = 2y-x+1, dy/dt = 3y-2x,
得 d^2y/dt^2 = 3dy/dt - 2dx/dt = 3dy/dt - 4y + 2x - 2
= 3dy/dt - 4y + 3y - dy/dt - 2 = 2dy/dt - y - 2
d^2y/dt^2 - 2dy/dt + y = -2
特征方程 r^2-2r+1 = 0, r = 1, 1
特解为 y = -2， 通解 y = (C1+C2t)e^t - 2
x = (1/2)(3y-dy/dt)
= (1/2)[(3C1+3C2t)e^t - 6 - (C1+C2+C2t)e^t]
= [(C1-C2/2+C2t)e^t - 3
微分方程组的解是
x = [(C1-C2/2+C2t)e^t - 3， y = (C1+C2t)e^t - 2

Answer 3

Answer 4

通解是：C&321;e^x+C&322;e^2x-（x^2+2x）e^x 解法：先求出齐次方程的通解，就是C1e^x+C2e^x 再求出一特解，齐次方程的通解+特解就是非齐次方程得解。特解的方法就是根据原方程等式右边的式子和齐次...