(x+1)(dy)/(dx)-x+4y=0

解

(x + 1) \frac{d y}{d x} - x + 4 y = 0

y = \frac{x ^{5}}{5 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{3 x ^{4}}{4 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{x ^{3}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} + \frac{x ^{2}}{2 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{c _{1}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} - \frac{1}{20 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )}

解答ステップ

(x + 1) \frac{d y}{d x} - x + 4 y = 0

線形 ODE を解く:

y = \frac{x ^{5}}{5 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{3 x ^{4}}{4 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{x ^{3}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} + \frac{x ^{2}}{2 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{c _{1}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} - \frac{1}{20 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )}

y = \frac{x ^{5}}{5 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{3 x ^{4}}{4 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{x ^{3}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} + \frac{x ^{2}}{2 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )} + \frac{c _{1}}{x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1} - \frac{1}{20 ( x ^{4} + 4 x ^{3} + 6 x ^{2} + 4 x + 1 )}