(dy)/((6+y)^2)=(dx)/((10+9x)^4)

해법

\frac{d y}{( 6 + y ) ^{2}} = \frac{d x}{( 10 + 9 x ) ^{4}}

y = - \frac{19683 x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{118098 c _{1} x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{65610 x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{393660 c _{1} x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{72900 x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{437400 c _{1} x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{162000 c _{1}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{26994}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}}

솔루션 단계

\frac{d y}{( 6 + y ) ^{2}} = \frac{d x}{( 10 + 9 x ) ^{4}}

분리 가능한 ODE 해결:

y = - \frac{19683 x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{118098 c _{1} x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{65610 x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{393660 c _{1} x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{72900 x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{437400 c _{1} x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{162000 c _{1}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{26994}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}}

y = - \frac{19683 x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{118098 c _{1} x ^{3}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{65610 x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{393660 c _{1} x ^{2}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{72900 x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{437400 c _{1} x}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{162000 c _{1}}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}} - \frac{26994}{- 1 + 19683 c _{1} x ^{3} + 65610 c _{1} x ^{2} + 72900 c _{1} x + 27000 c _{1}}