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We are given:

$$\begin{gathered} y\text{'}\text{'}\text{'}-y\text{'}-x=0 \\ \Rightarrow y\text{'}\text{'}\text{'}-y\text{'}=x \end{gathered}$$

Auxiliary equation is :

$$\begin{gathered} \left(m^3-m=0\right) \\ \Rightarrow m(m^2-1)=0 \\ \Rightarrow m=0, 1, -1 \end{gathered}$$

Hence, 

CF= Compliment Function = $c_1{e}^{0x}+c_2{e}^x+c_3{e}^{-x}=c_1+c_2{e}^x+c_3{e}^{-x}$

Now `will find the Particular Integral :

Here , F(x)=x

$$\begin{gathered} \Rightarrow P.I=\frac{1}{f\left(D\right)}F\left(x\right) \\ =\frac{1}{(D^3-D)}\left(x\right) \\ =\frac{1}{-D(1-D^2)}\left(x\right) \\ =\frac{-1}{D}{(1-D^2)}^{-1}\left(x\right) \\ =\frac{-1}{D}(1+D^2+D^4)\left(x\right) \\ =\frac{-1}{D}\left(x\right) \\ =-\frac{x^2}{2} \end{gathered}$$

Now, the General solution is = CF+PI= $c_1+c_2{e}^x+c_3{e}^{-x}-\frac{x^2}{2}$

Now, the General solution is = CF+PI= ${\mathit{c}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{x}}\mathbf{+}{\mathit{c}}_{\mathbf{3}}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathbf{-}\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{2}}$