Answer to Question #86129 in Electric Circuits for Ajay

Question #86129
Solve the differential equation: d^2y/dx^2+dy/dx+y=3sin3x
1
Expert's answer
2019-03-12T11:10:49-0400

Using Lagrange method (variation of constants),


"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"

"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"

"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."

"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)"

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_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))"


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