题目

Problem

Find the solution of the differential equation

x(x+7)\frac{\mathrm{d}y}{\mathrm{d}x}+7y=x

for which $y=7$ when $x=1$ . Give your answer in the form $y=\mathrm{f}(x)$ .

(9)

题目中文翻译

求微分方程

x(x+7)\frac{\mathrm{d}y}{\mathrm{d}x}+7y=x

满足当 $x=1$ 时 $y=7$ 的解。将答案写成 $y=\mathrm{f}(x)$ 的形式。

(9)

原卷通用作答要求

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。

解答

解法一

思路

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在含初值点的 $x>0$ 区间化成线性方程。 $y$ 的系数可拆成两个简单分式，找到积分因子后，右边再拆一次以便积分。

答题过程

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On $x>0$ , the equation is

y'+\frac7{x(x+7)}y=\frac1{x+7}.

Since $\frac7{x(x+7)}=\frac1x-\frac1{x+7}$ , the integrating factor is

\begin{align*} \mu=&\,e^{\int(1/x-1/(x+7))\,\mathrm{d}x}\\ =&\,e^{\ln x-\ln(x+7)}=\frac{x}{x+7}. \end{align*}

Multiplying by it gives

\frac{\mathrm{d}}{\mathrm{d}x}\bigg(\frac{xy}{x+7}\bigg) =\frac{x}{(x+7)^2} =\frac1{x+7}-\frac7{(x+7)^2}.

Integration gives

\begin{align*} \frac{xy}{x+7}=&\,\int\bigg(\frac1{x+7}-\frac7{(x+7)^2}\bigg)\,\mathrm{d}x\\ =&\,\ln(x+7)+\frac7{x+7}+C. \end{align*}

At $x=1$ , $y=7$ , so $7/8=\ln8+7/8+C$ , giving $C=-\ln8$ . Combining the logarithms and multiplying by $(x+7)/x$ gives

y=\frac{x+7}{x}\ln\bigg(\frac{x+7}{8}\bigg)+\frac7x.

CIE 9231 2022 June Paper 23 Q5

题目

解答

解法一

思路

答题过程