Using Lagrange method (variation of constants),
A ( x ) = − ∫ 1 W y 2 ( x ) 3 sin ( 3 x ) d x = − 2 3 ∫ e x 2 sin 3 x 2 sin 3 x d x A(x) = -\int \frac{1}{W} y_2(x) 3 \sin(3x)dx = -2\sqrt{3} \int e^{\frac{x}{2}} \sin{\frac{\sqrt{3}x}{2}}\sin{3x}dx A ( x ) = − ∫ W 1 y 2 ( x ) 3 sin ( 3 x ) d x = − 2 3 ∫ e 2 x sin 2 3 x sin 3 x d x
B ( x ) = ∫ 1 W y 1 ( x ) 3 sin ( 3 x ) d x = 2 3 ∫ e x 2 cos 3 x 2 sin 3 x d x B(x) = \int \frac{1}{W} y_1(x) 3 \sin(3x)dx = 2\sqrt{3} \int e^{\frac{x}{2}} \cos{\frac{\sqrt{3}x}{2}}\sin{3x}dx B ( x ) = ∫ W 1 y 1 ( x ) 3 sin ( 3 x ) d x = 2 3 ∫ e 2 x cos 2 3 x sin 3 x d x
A ( x ) = 3 e x / 2 ( ( 3 − 6 ) sin ( 1 2 ( 3 − 6 ) x ) + cos ( 1 2 ( 3 − 6 ) x ) 6 3 − 20 + ( 3 + 6 ) sin ( 1 2 ( 3 + 6 ) x ) + cos ( 1 2 ( 3 + 6 ) x ) 6 3 + 20 A(x) = \sqrt{3}e^{x/2}\left(\frac{\left(\sqrt{3}-6\right) \sin \left(\frac{1}{2} \left(\sqrt{3}-6\right) x\right)+\cos \left(\frac{1}{2} \left(\sqrt{3}-6\right) x\right)}{6 \sqrt{3}-20}+\frac{\left(\sqrt{3}+6\right) \sin \left(\frac{1}{2} \left(\sqrt{3}+6\right) x\right)+\cos \left(\frac{1}{2} \left(\sqrt{3}+6\right) x\right)}{6 \sqrt{3}+20}\right. A ( x ) = 3 e x /2 ( 6 3 − 20 ( 3 − 6 ) sin ( 2 1 ( 3 − 6 ) x ) + cos ( 2 1 ( 3 − 6 ) x ) + 6 3 + 20 ( 3 + 6 ) sin ( 2 1 ( 3 + 6 ) x ) + cos ( 2 1 ( 3 + 6 ) x )
B ( x ) = 3 e x / 2 ( sin ( 1 2 ( 3 + 6 ) x ) − ( 3 + 6 ) cos ( 1 2 ( 3 + 6 ) x ) 6 3 + 20 − sin ( 3 x − 3 x 2 ) + ( 3 − 6 ) cos ( 1 2 ( 3 − 6 ) x ) 6 3 − 20 ) B(x) = \sqrt{3} e^{x/2} \left(\frac{\sin \left(\frac{1}{2} \left(\sqrt{3}+6\right) x\right)-\left(\sqrt{3}+6\right) \cos \left(\frac{1}{2} \left(\sqrt{3}+6\right) x\right)}{6 \sqrt{3}+20}-\frac{\sin \left(3 x-\frac{\sqrt{3} x}{2}\right)+\left(\sqrt{3}-6\right) \cos \left(\frac{1}{2} \left(\sqrt{3}-6\right) x\right)}{6 \sqrt{3}-20}\right) B ( x ) = 3 e x /2 ⎝ ⎛ 6 3 + 20 sin ( 2 1 ( 3 + 6 ) x ) − ( 3 + 6 ) cos ( 2 1 ( 3 + 6 ) x ) − 6 3 − 20 sin ( 3 x − 2 3 x ) + ( 3 − 6 ) cos ( 2 1 ( 3 − 6 ) x ) ⎠ ⎞ Substituting A(x) and B(x) to the general solution of the homogeneous eq. we get general solution of the nonhomogeneous eq.:
y ( x ) = c 1 e − x 2 sin ( 3 x 2 ) + c 2 e − x 2 cos ( 3 x 2 ) − 3 73 ( 8 sin ( 3 x ) + 3 cos ( 3 x ) ) y(x)= c_1e^{-\frac{x}{2}} \sin \left(\frac{\sqrt{3} x}{2}\right)+c_2 e^{-\frac{x}{2}} \cos \left(\frac{\sqrt{3} x}{2}\right)-\frac{3}{73} (8 \sin (3 x)+3 \cos (3 x)) y ( x ) = c 1 e − 2 x sin ( 2 3 x ) + c 2 e − 2 x cos ( 2 3 x ) − 73 3 ( 8 sin ( 3 x ) + 3 cos ( 3 x ))
#339914 on Dec 2023
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