求微分方程

满足当 $x=0$ 时，$y=0$ 且 $\dfrac{\mathrm{d}y}{\mathrm{d}x}=0$ 的特解。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

先解辅助方程。右端是二次多项式，试同次数的特解。最后用 $y(0)$ 和 $y'(0)$ 确定互补函数中的常数。

The auxiliary equation $5m^2+4m+1=0$ gives $m=-\frac25\pm\frac15\mathrm{i}$. The complementary function is

For the trial $y_p=px^2+qx+r$, the derivatives are $y_p'=2px+q$ and $y_p''=2p$. Substitution gives $(p)x^2+(8p+q)x+(10p+4q+r)=x^2+7x+1$. Comparing coefficients,

Thus $q=-1$ and $r=-5$. The general solution and its derivative are

Applying the two initial conditions gives

So $A=5$ and $B=15$, giving