$\because y=c_1 e^x+c_2 e^{\log c x}+c_3 \sin ^2 x-c_4\left(\cos ^2 x-1\right)$
$\Rightarrow y=c_1 e^x+c_2 x+\left(c_3+c_4\right) \sin ^2 x$
$\Rightarrow y=c_1 e^x+c_2 x+c_3^{\prime} \sin ^2 x$ ...(i)
where $c_3^{\prime}=c_3+c_4$
$y^{\prime}=c_1 e^x+c_2+c_3^{\prime} \sin 2 x$ ...(ii)
$y^{\prime \prime}=c_1 e^x+c_2 \cdot 0+2 c_3^{\prime} \cos 2 x$ ...(iii)
$\Rightarrow y^{\prime \prime \prime}=c_1 e^x+c_2 \cdot 0+c_3^{\prime}(-4 \sin 2 x)$ ...(iv)
Eliminating $c_1, c_2 \& c_3$ from equation (i), (ii), (iii) \& (iv), we get
$\left|\begin{array}{cccc}y & -e^x & -x & -\sin ^2 x \\ y^{\prime} & -e^x & -1 & -\sin 2 x \\ y^{\prime \prime} & -e^x & 0 & -2 \cos 2 x \\ y^{\prime \prime} & -e^x & 0 & +4 \sin 2 x\end{array}\right|=0$
Clearly, after solving above determinant, we will get a differential equation with order 3 .