Question

求微分方程特解形式

求微分方程特解形式
求到y*之后带入不会算了
麻烦详细写一下步骤

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Answer 1

特征方程 r^2+6r+9 = 0, r = -3，-3
特解形式 y = x^2(ax+b)e^(-3x) = (ax^3+bx^2)e^(-3x)
y' = (3ax^2+2bx)e^(-3x) - 3(ax^3+bx^2)e^(-3x)
= [-3ax^3+(3a-3b)x^2+2bx]e^(-3x)
y'' = [-9ax^2+2(3a-3b)x+2b]e^(-3x) - 3[-3ax^3+(3a-3b)x^2+2bx]e^(-3x)
= [9ax^3+(-18a+9b)x^2+(6a-12b)x+2b]e^(-3x)
代入微分方程得
[9ax^3+(-18a+9b)x^2+(6a-12b)x+2b] + 6[-3ax^3+(3a-3b)x^2+2bx] + 9(ax^3+bx^2) = 3x
2b = 0, b = 0;
6a = 3, a = 1/2
特解 y = (1/2)x^3 e^(-3x)

Answer 2

特解+通解就是结果