高数微分方程
5y^2-2yy'=4cosx+3sinx,
d(y^2)/dx-5y^2=-(4cosx+3sinx), 为y^2对x的一阶线性方程,则
y^2=e^(∫5dx)[-∫(4cosx+3sinx)e^(-∫5dx)dx+C]
=e^(5x)[-∫(4cosx+3sinx)e^(-5x)dx+C],
其中 -∫(4cosx+3sinx)e^(-5x)dx=(1/5)∫(4cosx+3sinx)de^(-5x)
=(1/5)(4cosx+3sinx)e^(-5x)-(1/5)∫(-4sinx+3cosx)e^(-5x)dx
=(1/5)(4cosx+3sinx)e^(-5x)+(1/25)∫(...全部
5y^2-2yy'=4cosx+3sinx,
d(y^2)/dx-5y^2=-(4cosx+3sinx), 为y^2对x的一阶线性方程,则
y^2=e^(∫5dx)[-∫(4cosx+3sinx)e^(-∫5dx)dx+C]
=e^(5x)[-∫(4cosx+3sinx)e^(-5x)dx+C],
其中 -∫(4cosx+3sinx)e^(-5x)dx=(1/5)∫(4cosx+3sinx)de^(-5x)
=(1/5)(4cosx+3sinx)e^(-5x)-(1/5)∫(-4sinx+3cosx)e^(-5x)dx
=(1/5)(4cosx+3sinx)e^(-5x)+(1/25)∫(-4sinx+3cosx)de^(-5x)
=(1/5)(4cosx+3sinx)e^(-5x)+(1/25)(-4sinx+3cosx)e^(-5x)
+(1/25)∫(4cosx+3sinx)e^(-5x)dx,
则(-26/25)∫(4cosx+3sinx)e^(-5x)dx=(1/25)(23cosx+11sinx)e^(-5x),
-∫(4cosx+3sinx)e^(-5x)dx=(1/26)(23cosx+11sinx)e^(-5x)。
于是 y^2=e^(5x)[(1/26)(23cosx+11sinx)e^(-5x)+C]
=(1/26)(23cosx+11sinx)+Ce^(5x)。收起