V^'=4kpi(3/(4pi))^{2/3}V^{2/3}

解

V^{^{'}} = 4 kπ (\frac{3}{4 π})^{\frac{2}{3}} V^{\frac{2}{3}}

V (t) = \frac{4 π k ^{3} t ^{3}}{3} + \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}} c _{1} k ^{2} t ^{2}}{3 ^{\frac{2}{3}}} + \frac{4 ^{\frac{1}{3}} π ^{\frac{1}{3}} c _{1}^{2} k t}{3 \cdot 3 ^{\frac{1}{3}}} + \frac{c _{1}^{3}}{27}

解答ステップ

V^{'} (t) = 4 kπ (\frac{3}{4 π})^{\frac{2}{3}} V (t)^{\frac{2}{3}}

分離可能 ODE を解く:

V (t) = \frac{4 π k ^{3} t ^{3}}{3} + \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}} c _{1} k ^{2} t ^{2}}{3 ^{\frac{2}{3}}} + \frac{4 ^{\frac{1}{3}} π ^{\frac{1}{3}} c _{1}^{2} k t}{3 \cdot 3 ^{\frac{1}{3}}} + \frac{c _{1}^{3}}{27}

V (t) = \frac{4 π k ^{3} t ^{3}}{3} + \frac{4 ^{\frac{2}{3}} π ^{\frac{2}{3}} c _{1} k ^{2} t ^{2}}{3 ^{\frac{2}{3}}} + \frac{4 ^{\frac{1}{3}} π ^{\frac{1}{3}} c _{1}^{2} k t}{3 \cdot 3 ^{\frac{1}{3}}} + \frac{c _{1}^{3}}{27}