\(\frac{d^2y}{d\mathrm x^2}+\frac{dy}{d\mathrm x}+y\) \(=(1-e^\mathrm x)^2\)
Its auxiliary equation is
m2 + m + 1 = 0
\(\Rightarrow m=\frac{-1\pm \sqrt{1-4}}{2}\)
\(=\frac{-1}{2}\pm \frac{\sqrt 3}{2} i\)
Therefore, C.F. \(=e^{\frac{-\mathrm x}{2}}\)\((c_1\cos\left(\frac{\sqrt 3}{2}\mathrm x\right)\)+ \(c_2\sin \left(\frac{\sqrt 3}{2}\mathrm x\right))\)
P.I \(=\frac{1}{D^2+D+1}\)(1 – ex)2
\(=\frac{1}{D^2+D+1}(e^{2\mathrm x}-2e^{\mathrm x}+e^o)\) (∵ eo = 1)
\(=\frac{e^{2\mathrm x}}{2^2+2+1}-\frac{2e^\mathrm x}{1^2+1+1}+\frac{1}{0+0+1}\)
\(=\frac{e^{2\mathrm x}}{7}-\frac{2e^\mathrm x}{3}+1\)
Hence, solution of given differential equation is
y = C.F + P.I
\(=e^{-\frac{\mathrm x}{2}}(c_1\cos\left(\frac{\sqrt 3}{2}\mathrm x\right)\) + \(c_2\sin \left(\frac{\sqrt 3}{2}\mathrm x\right))\) + \(\frac{e^{2\mathrm x}}{7}-\frac{2e^\mathrm x}{3}+1\)