(dy)/(dx)= 2/3 (x^2+1)^{3/2}

解

\frac{d y}{d x} = \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}

y = \frac{1}{12} (2 x (x^{2} + 1)^{\frac{3}{2}} + 3 ln (x + x^{2} + 1) + 3 x x^{2} + 1) + c_{1}

解答ステップ

\frac{d y}{d x} = \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}

代用

\frac{d y}{d x}

と

y^{'} (x)

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

f^{^{'}} (x) = g (x)

であれば

f (x) = \int g (x) d x

y = \int \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}} d x

\int \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}} d x = \frac{1}{12} (2 x (x^{2} + 1)^{\frac{3}{2}} + 3 ln (x + x^{2} + 1) + 3 x x^{2} + 1) + c_{1}

y = \frac{1}{12} (2 x (x^{2} + 1)^{\frac{3}{2}} + 3 ln (x + x^{2} + 1) + 3 x x^{2} + 1) + c_{1}