设y=f(x)是微分方程y''+2y'+3y=e^3x满足初值条件y(0)=y'(0)=0的特解,求极限lim(x->0)[ln(1+x^2)]/f(x)

Question

设y=f(x)是微分方程y''+2y'+3y=e^3x满足初值条件y(0)=y'(0)=0的特解,求极限lim(x->0) [ln(1+x^2)]/f(x)

Answer 1

∵y=f(x)是微分方程y''+2y'+3y=e^3x满足初值条件y(0)=y'(0)=0的特解,
∴y''=e^3x-2y'-3y.
故极限lim(x->0) [ln(1+x^2)]/f(x)
=lim(x->0)[2x/(1+x^2)]/f'(x)
=2lim(x->0)x/f'(x)
=2lim(x->0)1/f''(x)
=2lim(x->0)1/[e^3x-2y'-3y]
=2.