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3.7k views
in Differential Equations by (69 points)
edited by

Solve d2y/dx2+dy/dx+y=(1-ex)2

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1 Answer

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by (710 points)

\(\frac{d^2y}{d\mathrm x^2}+\frac{dy}{d\mathrm x}+y\) \(=(1-e^\mathrm x)^2\)

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

\(=e^{-\frac{\mathrm x}{2}}(c_1\cos\left(\frac{\sqrt 3}{2}\mathrm x\right)\) + \(c_2\sin \left(\frac{\sqrt 3}{2}\mathrm x\right))\) + \(\frac{e^{2\mathrm x}}{7}-\frac{2e^\mathrm x}{3}+1\)

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