y^'=5t^3y^{1/5}

解答

y^{^{'}} = 5 t^{3} y^{\frac{1}{5}}

y = \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}, y = - \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}

求解步骤

y^{'} (t) = 5 t^{3} y^{\frac{1}{5}}

求解可分离方程:

y = \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}, y = - \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}

y = \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}, y = - \frac{4 3125 t ^{20} + 12500 c _{1} t ^{16} + 20000 c _{1}^{2} t ^{12} + 16000 c _{1}^{3} t ^{8} + 6400 c _{1}^{4} t ^{4} + 1024 c _{1}^{5}}{5 4 5}

y^{^{'}} = \frac{3 x ^{2} + 4 x + 2}{2 ( y - 1 )}

\frac{d y}{d x} + 2 y = 6

y^{^{'}} (x) = \frac{1 + 3 x ^{2}}{3 y ^{2} - 6 y}, y (0) = 1

(x^{2} y + 4 x y + 2 y) d x + (x^{2} + x) d y = 0

x y^{3} d x + (1 - x) d y = 0, y (2) = 1