( D 2 − 2 D + 4 ) y = 8 x 2 e 2 x sin 2 x The solution to the above equation is y = y c + y p where y c is the complementary factor and y p is the particular integral The auxiliary equation to this ODE is m 2 − 2 m + 4 = 0 m = 2 ± 4 − 16 2 = 2 ± j 12 2 = 2 ± j 2 3 2 = 1 ± j 3 The complementary solution to this ODE is y = e x ( A cos ( 3 x ) + B sin ( 3 ) ) The Wronskian of the two solutions is W ( x ) = e − ∫ − 2 d x = e 2 x ∴ Our particular solution will be given by y p = V 1 ( x ) e x sin ( 3 x ) + V 2 ( x ) e x cos ( 3 x ) Where V 1 ( x ) = − ∫ r ( x ) sin ( 3 x ) W ( x ) d x , V 2 ( x ) = ∫ r ( x ) cos ( 3 x ) W ( x ) d x and V 1 ( x ) = − ∫ 8 x 2 e 2 x sin 2 x sin ( 3 x ) e 2 x d x = − ∫ 8 x 2 sin 2 x sin ( 3 x ) d x = − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( ( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x ) ) − 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) cos ( 3 x ) + C V 2 ( x ) = ∫ 8 x 2 e 2 x sin 2 x cos ( 3 x ) e 2 x d x = ∫ 8 x 2 sin 2 x cos ( 3 x ) d x = 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x ) ) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x ) ) + C y p = e x sin ( 3 x ) ( − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( ( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x ) ) − 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) cos ( 3 x ) ) + e x cos ( 3 x ) ( 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x ) ) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x ) ) ∴ y = e x ( A cos ( 3 x ) + B sin ( 3 ) ) + e x sin ( 3 x ) ( − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( ( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x ) ) − 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) cos ( 3 x ) ) + e x cos ( 3 x ) ( 16 ( ( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x ) ) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x ) ) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x ) ) (D^2 - 2D + 4)y = 8x^2e^{2x}\sin{2x}
\\ \textsf{The solution to the above equation is}
\\ y = y_c + y_p \\
\textsf{where}\, y_c \, \textsf{is the complementary factor and}
\\ y_p\, \textsf{ is the particular integral}
\textsf{The auxiliary equation to this ODE is}\\
\begin{aligned}
m^2 - 2m + 4 &= 0\\
m &= \frac{2 \pm \sqrt{4 - 16}}{2} = \frac{2 \pm j\sqrt{12}}{2}
\\&=\frac{2 \pm j2\sqrt{3}}{2}= 1 \pm j\sqrt{3}
\end{aligned}\\
\textsf{The complementary solution to this ODE is}\\
y = e^{x}(A\cos(\sqrt{3}x) + B\sin(\sqrt{3}))\\
\textsf{The Wronskian of the two solutions is} \\
W(x) = e^{-\int-2\mathrm{d}x} = e^{2x}\\
\therefore\textsf{ Our particular solution will be given by}\\
y_p = V_1(x)e^{x}\sin(\sqrt{3}x) + V_2(x)e^{x}\cos(\sqrt{3}x)\\
\textsf{Where}\, V_1(x) = -\int \frac{r(x)\sin(\sqrt{3}x)}{W(x)} \, \mathrm{d}x,\, V_2(x) = \int \frac{r(x)\cos(\sqrt{3}x)}{W(x)}\, \mathrm{d}x \, \textsf{and} \, \\
\begin{aligned}
V_1(x) &= -\int \frac{8x^2e^{2x}\sin{2x} \sin(\sqrt{3}x)}{e^{2x}} \, \mathrm{d}x\\
&= -\int 8x^2\sin{2x} \sin(\sqrt{3}x) \, \mathrm{d}x\\
&= - 8\sqrt{3} (8 x \sin^2(x) - (x^2 - 30) \sin(2 x)) \sin(\sqrt{3} x) \\
& + 16 \cos^2(x) ((x^2 - 26) \cos(\sqrt{3} x) + 4 \sqrt{3} x \sin(\sqrt{3} x)) \\
&- 16 ((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \cos(\sqrt{3} x) + C
\end{aligned}\\
\begin{aligned}
V_2(x) &= \int \frac{8x^2e^{2x}\sin{2x} \cos(\sqrt{3}x)}{e^{2x}} \, \mathrm{d}x\\
&= \int 8x^2\sin{2x} \cos(\sqrt{3}x) \, \mathrm{d}x\\
&= 16((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \sin(\sqrt{3} x) \\
&+ 16\cos^2(x) (4 \sqrt{3} x \cos(\sqrt{3} x) - (x^2 - 26) \sin(\sqrt{3} x)) \\
&+ 16\sqrt{3} (x^2 - 30) \sin(x) \cos(\sqrt{3} x) \cos(x) \\
&- 64 \sqrt{3} x \sin^2(x) \cos(\sqrt{3} x)) + C
\end{aligned}\\
\begin{aligned}
y_p &= e^{x}\sin(\sqrt{3}x)(- 8\sqrt{3} (8 x \sin^2(x) - (x^2 - 30) \sin(2 x)) \sin(\sqrt{3} x) \\
& + 16 \cos^2(x) ((x^2 - 26) \cos(\sqrt{3} x) + 4 \sqrt{3} x \sin(\sqrt{3} x)) \\
&- 16 ((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \cos(\sqrt{3} x)) \\
&+ e^{x}\cos(\sqrt{3}x)(16((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \sin(\sqrt{3} x) \\
&+ 16\cos^2(x) (4 \sqrt{3} x \cos(\sqrt{3} x) - (x^2 - 26) \sin(\sqrt{3} x)) \\
&+ 16\sqrt{3} (x^2 - 30) \sin(x) \cos(\sqrt{3} x) \cos(x) \\
&- 64 \sqrt{3} x \sin^2(x) \cos(\sqrt{3} x))
\end{aligned}\\
\begin{aligned}
\therefore y &= e^{x}(A\cos(\sqrt{3}x) + B\sin(\sqrt{3}))
\\&+ e^{x}\sin(\sqrt{3}x)(- 8\sqrt{3} (8 x \sin^2(x) - (x^2 - 30) \sin(2 x)) \sin(\sqrt{3} x) \\
& + 16 \cos^2(x) ((x^2 - 26) \cos(\sqrt{3} x) + 4 \sqrt{3} x \sin(\sqrt{3} x)) \\
&- 16 ((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \cos(\sqrt{3} x)) \\
&+ e^{x}\cos(\sqrt{3}x)(16((x^2 - 26) \sin^2(x) + 7 x \sin(2 x)) \sin(\sqrt{3} x) \\
&+ 16\cos^2(x) (4 \sqrt{3} x \cos(\sqrt{3} x) - (x^2 - 26) \sin(\sqrt{3} x)) \\
&+ 16\sqrt{3} (x^2 - 30) \sin(x) \cos(\sqrt{3} x) \cos(x) \\
&- 64 \sqrt{3} x \sin^2(x) \cos(\sqrt{3} x))
\end{aligned}\\ ( D 2 − 2 D + 4 ) y = 8 x 2 e 2 x sin 2 x The solution to the above equation is y = y c + y p where y c is the complementary factor and y p is the particular integral The auxiliary equation to this ODE is m 2 − 2 m + 4 m = 0 = 2 2 ± 4 − 16 = 2 2 ± j 12 = 2 2 ± j 2 3 = 1 ± j 3 The complementary solution to this ODE is y = e x ( A cos ( 3 x ) + B sin ( 3 )) The Wronskian of the two solutions is W ( x ) = e − ∫ − 2 d x = e 2 x ∴ Our particular solution will be given by y p = V 1 ( x ) e x sin ( 3 x ) + V 2 ( x ) e x cos ( 3 x ) Where V 1 ( x ) = − ∫ W ( x ) r ( x ) s i n ( 3 x ) d x , V 2 ( x ) = ∫ W ( x ) r ( x ) c o s ( 3 x ) d x and V 1 ( x ) = − ∫ e 2 x 8 x 2 e 2 x sin 2 x sin ( 3 x ) d x = − ∫ 8 x 2 sin 2 x sin ( 3 x ) d x = − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) (( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x )) − 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) cos ( 3 x ) + C V 2 ( x ) = ∫ e 2 x 8 x 2 e 2 x sin 2 x cos ( 3 x ) d x = ∫ 8 x 2 sin 2 x cos ( 3 x ) d x = 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x )) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x )) + C y p = e x sin ( 3 x ) ( − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) (( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x )) − 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) cos ( 3 x )) + e x cos ( 3 x ) ( 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x )) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x )) ∴ y = e x ( A cos ( 3 x ) + B sin ( 3 )) + e x sin ( 3 x ) ( − 8 3 ( 8 x sin 2 ( x ) − ( x 2 − 30 ) sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) (( x 2 − 26 ) cos ( 3 x ) + 4 3 x sin ( 3 x )) − 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) cos ( 3 x )) + e x cos ( 3 x ) ( 16 (( x 2 − 26 ) sin 2 ( x ) + 7 x sin ( 2 x )) sin ( 3 x ) + 16 cos 2 ( x ) ( 4 3 x cos ( 3 x ) − ( x 2 − 26 ) sin ( 3 x )) + 16 3 ( x 2 − 30 ) sin ( x ) cos ( 3 x ) cos ( x ) − 64 3 x sin 2 ( x ) cos ( 3 x ))
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