常见辅助函数

$f^{\prime \prime }(x)+f(x)$

辅助函数：
$F(x)=[f(x){]}^2+[f^{\prime }(x){]}^2$

$f^{\prime \prime }(x)f(x)+[f^{\prime }(x){]}^2$

辅助函数：
$$\begin{gathered} F(x)=f^{\prime }(x)f(x) \\ {F}^{\prime }(x)=f^{\prime \prime }(x)f(x)+[f^{\prime }(x){]}^2 \end{gathered}$$

$f^{\prime \prime }(x)f(x)-[f^{\prime }(x){]}^2$

辅助函数：
$$\begin{gathered} F(x)=\frac{f^{\prime }(x)}{f(x)} \\ {F}^{\prime }(x)=\frac{f^{\prime \prime }(x)f(x)-[f^{\prime }(x){]}^2}{f^2(x)} \end{gathered}$$

$f^{\prime }(x)+xf(x)$

辅助函数：
$$\begin{gathered} F(x)=f(x)e^{\frac{x^2}{2}} \\ {F}^{\prime }(x)=[f^{\prime }(x)+xf(x)]e^{\frac{x^2}{2}} \end{gathered}$$
拓展：$$\begin{gathered} f^{\prime }(x)+g(x)f(x) \\ 辅助函数：F(x)=f(x)\cdot e^{\int g(x)dx} \\ {F}^{\prime }(x)=[f^{\prime }(x)+g(x)f(x)]e^{\int g(x)dx} \end{gathered}$$

$f^{\prime }(x)-xf(x)$

辅助函数：
$$\begin{gathered} F(x)=\frac{f(x)}{e^{\frac{x^2}{2}}} \\ {F}^{\prime }(x)=\frac{[f^{\prime }(x)-xf(x)]}{e^{\frac{x^2}{2}}} \end{gathered}$$

或者

$$\begin{gathered} F(x)=f(x)e^{\frac{-x^2}{2}} \\ {F}^{\prime }(x)=[f^{\prime }(x)-xf(x)]e^{\frac{-x^2}{2}} \end{gathered}$$

$f^{\prime }(x)-2f(x)tanx$

辅助函数：
$$\begin{gathered} F(x)=f(x)co{s}^{2}x \\ {F}^{\prime }(x)=f^{\prime }(x)-2f(x)tanx \end{gathered}$$