$\begin{array} { l }P\left( x \right)=2x,\\Q\left( x \right)=x\end{array}$
Insert the function $P\left( x \right)=2x$ into the formula for the integrating factor $u\left( x \right)$$\begin{array} { l }u\left( x \right)={e}^{\int{ 2x } \mathrm{d} x},\\Q\left( x \right)=x\end{array}$
Evaluate the integral$\begin{array} { l }u\left( x \right)={e}^{\left( {x}^{2} \right)},\\Q\left( x \right)=x\end{array}$
Insert the integrating factor $u\left( x \right)$ and the function $Q\left( x \right)$ into the general solution formula$y=\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} } \times \int{ x{e}^{\left( {x}^{2} \right)} } \mathrm{d} x$
Evaluate the integral$\begin{array} { l }y=\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} } \times \left( \frac{ {e}^{\left( {x}^{2} \right)} }{ 2 }+C \right),& C \in ℝ\end{array}$
Multiply each term in the parentheses by $\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} }$$\begin{array} { l }y=\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} } \times \frac{ {e}^{\left( {x}^{2} \right)} }{ 2 }+\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} } \times C,& C \in ℝ\end{array}$
Cancel out the common factor ${e}^{\left( {x}^{2} \right)}$$\begin{array} { l }y=\frac{ 1 }{ 2 }+\frac{ 1 }{ {e}^{\left( {x}^{2} \right)} } \times C,& C \in ℝ\end{array}$
Calculate the product$\begin{array} { l }y=\frac{ 1 }{ 2 }+\frac{ C }{ {e}^{\left( {x}^{2} \right)} },& C \in ℝ\end{array}$