Question

y'+(2-3x^2)/x^3y=1,y|x=1=0微分方程的特解

Answer 1

Answer 2

y(x) = (1/2)*x^3*(1+exp(1/x^2-1))

Answer 3

一阶微分方程的公式法。

Answer 4

y(x) = (1/2)*x^3*(1+exp(1/x^2-1))

Answer 5

将方程改为dy/dx+(2-3x²)/x³y=1
其对应的齐次方程为dy/dx+(2-3x²)/x³=0
通解:y=Ce^-∫(2-3x²)/x³dx=Ce^(1/x²)x³
令非齐次方程通解为y=C(x)e^(1/x²)x³
代入非齐次方程得C'(x)e^(1/x²)x³=1
积分得C(x)=1/2e^(-1/x²)+C
故原方程通解为y=1/2x³+Cx³e^(1/x²)
有初值条件得C=-1/2e^(-1)
故特解为y=1/2x³-1/2x³e^(1/x²-1)