D2y + 2y = x2e3x + ex cos 2x.

1.	D2y + 2y = x2e3x + ex cos 2x.
Answer» It's auxiliary equation is m² +2 = 0 m = \(\pm\sqrt2i\) ∴ C.F. = C₁ cos\(\sqrt2\)x + C₂sin √2x P. I. = \(\frac1{D^2+2}x^2e^{3x}+\frac1{D^2+2}e^xcos2x\) = e^3x \(\frac1{(D+3)^2+2}x^2+e^x\frac1{(D+1)^2+2}cos 2x\) = e^3x\(\frac1{11+6D+D^2}x^2+e^x\frac{1}{D^2+2D+3}cos 2x\) = \(\frac{e^{3x}}{11}(1+\frac{6D+D^2}{11})^{-1}\)x² + e^x \(\frac1{-4+2d+3}cos 2x\) = \(\frac{e^{3x}}{11}(1-\frac{6D+D^2}{11}+\frac{36}{121}D^2+....)x^2\) + e^x\(\frac{2D+1}{4D^2-1}cos 2x\) = \(\frac{e^{3x}}{1331}(121x^2-132x + 50)-\frac{e^x}{17}(cos 2x - 4sin 2x)\) ∴ y = C. F. + P. I. = C₁cos(√2x) + C₂sin(√2x) + \(\frac{e^{3x}}{1331}\)(121x² - 132x + 50) - \(\frac{e^x}{17}\)(cos 2x - 4 sin 2x)

Answer»

It's auxiliary equation is

m² +2 = 0

m = \(\pm\sqrt2i\)

∴ C.F. = C₁ cos\(\sqrt2\)x + C₂sin √2x

P. I. = \(\frac1{D^2+2}x^2e^{3x}+\frac1{D^2+2}e^xcos2x\)

= e^3x \(\frac1{(D+3)^2+2}x^2+e^x\frac1{(D+1)^2+2}cos 2x\)

= e^3x\(\frac1{11+6D+D^2}x^2+e^x\frac{1}{D^2+2D+3}cos 2x\)

= \(\frac{e^{3x}}{11}(1+\frac{6D+D^2}{11})^{-1}\)x² + e^x \(\frac1{-4+2d+3}cos 2x\)

= \(\frac{e^{3x}}{11}(1-\frac{6D+D^2}{11}+\frac{36}{121}D^2+....)x^2\) + e^x\(\frac{2D+1}{4D^2-1}cos 2x\)

= \(\frac{e^{3x}}{1331}(121x^2-132x + 50)-\frac{e^x}{17}(cos 2x - 4sin 2x)\)

∴ y = C. F. + P. I.

= C₁cos(√2x) + C₂sin(√2x) + \(\frac{e^{3x}}{1331}\)(121x² - 132x + 50) - \(\frac{e^x}{17}\)(cos 2x - 4 sin 2x)

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