y^'+7/(x+3)y=(x+3)^3

解

y^{^{'}} + \frac{7}{x + 3} y = (x + 3)^{3}

y = \frac{x ^{11}}{11 ( x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187 )} + \frac{3 x ^{10}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{45 x ^{9}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{405 x ^{8}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{2430 x ^{7}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{10206 x ^{6}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{30618 x ^{5}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{65610 x ^{4}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{3}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{2}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{59049 x}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{c _{1}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

解答ステップ

y^{'} (x) + \frac{7}{x + 3} y = (x + 3)^{3}

線形 ODE を解く:

y = \frac{x ^{11}}{11 ( x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187 )} + \frac{3 x ^{10}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{45 x ^{9}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{405 x ^{8}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{2430 x ^{7}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{10206 x ^{6}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{30618 x ^{5}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{65610 x ^{4}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{3}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{2}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{59049 x}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{c _{1}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}

y = \frac{x ^{11}}{11 ( x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187 )} + \frac{3 x ^{10}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{45 x ^{9}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{405 x ^{8}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{2430 x ^{7}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{10206 x ^{6}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{30618 x ^{5}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{65610 x ^{4}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{3}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{98415 x ^{2}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{59049 x}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187} + \frac{c _{1}}{x ^{7} + 21 x ^{6} + 189 x ^{5} + 945 x ^{4} + 2835 x ^{3} + 5103 x ^{2} + 5103 x + 2187}