\[ 2y^{\prime \prime }-e^{y}=0 \] Multiplying both sides by \(y^{\prime }\) gives\[ 2y^{\prime }y^{\prime \prime }-y^{\prime }e^{y}=0 \] Integrating\begin {align*} \int \left ( 2y^{\prime }y^{\prime \prime }-y^{\prime }e^{y}\right ) dx & =c_{1}\\ \left ( y^{\prime }\right ) ^{2}-e^{y} & =c_{1} \end {align*}
Hence\[ y^{\prime }=\pm \sqrt {e^{y}+c_{1}}\] Each of the above is separable, which are solved by integration.