浙江大学真题 第21000题
📝 题目
三（20分）设 $\varphi(t)$ 和 $\psi(t)$ 为二次可微函数，
$$
u(x, y)=x \varphi(x+y)+y \psi(x+y)
$$
证明 $\frac{\partial^{2} u}{\partial x^{2}}-2 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=0$
证
$$
\begin{aligned}
& u_{x}=\varphi+x \varphi^{\prime}+y \psi^{\prime}, \quad u_{y}=x \varphi^{\prime}+\psi+y \psi^{\prime} \\
& u_{x x}=2 \varphi^{\prime}+x \varphi^{\prime \prime}+y \psi^{\prime \prime}, \quad u_{x y}=\varphi^{\prime}+x \varphi^{\prime \prime}+\psi^{\prime}+y \psi^{\prime \prime}
\end{aligned}
$$
$$
\begin{equation*}
u_{y y}=x \varphi^{\prime \prime}+2 \psi^{\prime}+y \psi^{\prime \prime} \tag{15 分}
\end{equation*}
$$
因此，左 $=2 \varphi^{\prime}+x \varphi^{\prime \prime}+y \psi^{\prime \prime}-2\left(\varphi^{\prime}+x \varphi^{\prime \prime}+\psi^{\prime}+y \psi^{\prime \prime}\right)$
$$
\begin{equation*}
+x \varphi^{\prime \prime}+2 \psi^{\prime}+y \psi^{\prime \prime}=0=\text { 右 } \tag{20 分}
\end{equation*}
$$
💡 答案解析
📋 详细解题步骤
步骤 1/1
目标：计算一阶偏导，再计算二阶偏导，代入验证二阶偏导组合为零。
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