Symon2-53

l=53\n0<l< 2\\pi\nm \\frac{d^2x}{dt^2} + kx = 0 \\, x(0) = 0, \\, V(0) = V_0\n\\frac{dx}{dt} + kx = 0\nx'' + \\frac{k}{m}x = 0\np + \\omega^2 = 0\np = -\\omega^2\nX = C_1 \\cos(\\omega t) + C_2 \\sin(\\omega t)\n\\alpha = \\omega (t)\nV(x) = V_0 C_2\nd= C_2\\frac{V_0}{\\omega}\nX = \\frac{V_0}{\\omega} \\sin(\\omega t), \\, 0 < t < \\frac{2\\pi}{\\omega}\n\nt= \\frac{3}{2\\omega}\nm \\frac{d^2x}{dt^2} + kx = B \\cos(\\omega t + \\theta)\nx'' + \\frac{k}{m}x = \\frac{B}{m} \\cos(\\omega t + \\theta)\nx'' + \\frac{V_0}{k} = \\frac{V_0}{k} \\cos(\\omega t)\nx_h = K_1 \\cos(\\omega t) + K_2 \\sin(\\omega t)\nx_p = C_3 \\cos(\\omega t + \\theta) + C_4 \\sin(\\omega t + \\theta)\n\\frac{dx}{dt} = K_1 \\omega \\sin(\\omega t) + K_2 \\omega \\cos(\\omega t)\n\ndx = 0 = C_3 \\cos(\\theta) - K_2 \\omega sin(\\omega t + \\theta)\n-k_3 \\cos(\\omega t + \\theta) - \\omega^2 C_3 \\sin(\\omega t + \\theta \n\n-k_3 \\cos(\\theta) - \\omega^2\n\\frac{k_2 B}{m} \\cos(\\omega t + \\theta) = \\frac{B}{m} \\cos(\\omega t + \\theta)\\nC_3 \\left( \\frac{\\omega^{2}}{m} - \\omega^{2} \\right) = 0\\nC_4 = 0 X = x_h + x_p\nX = K_1 \\cos(\\omega t) + K_2 \\sin(\\omega t) + \\frac{B}{m(\\omega^{2} - \\omega_{0}^{2})} \\cos(\\omega t + \\theta)\nV_0 = \\left(\\frac{V_0}{\\omega} \\right) \\left(\\frac{3}{2\\omega}\\right) - K_2 \\left(\\frac{V_0}{\\omega^{4+p}} \\right) \\cos(\\theta)\n\\alpha = \\frac{\\pi}{3} + \\theta\\nK_2 = \\frac{V_2}{\\omega c}\n\\frac{dx}{dt} = -\\omega K_1 \\sin(\\omega t) + K_2 \\omega \\cos(\\omega t)\nX = K_1 \\sin(\\omega t) + K_2 \\sin(\\omega t)\nX = \\frac{\\sqrt{\\frac{V_0}{m(\\omega^{2}-\\omega_{0}^{2})}}}{K_1} \\sin(\\omega t) + K_2 \\cos(\\omega t)\nt2\n X = x_h + x_p