$$ \frac { d y } { d t } = y ( y - 1 ) ^ { 2 } ( y - 2 ) ^ { 3 } ( y - 3 ) ^ { 4 } $$

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To solve the differential equation $$ \frac{dy}{dt} = y (y - 1)^2 (y - 2)^3 (y - 3)^4, $$ we can follow these steps: ### 1. **Separate Variables** First, separate the variables $$ y $$ and $$ t $$: $$ \frac{dy}{y (y - 1)^2 (y - 2)^3 (y - 3)^4} = dt $$ ### 2. **Integrate Both Sides** Integrate both sides to find $$ t $$ as a function of $$ y $$: $$ \int \frac{1}{y (y - 1)^2 (y - 2)^3 (y - 3)^4} \, dy = \int dt = t + C $$ where $$ C $$ is the constant of integration. ### 3. **Partial Fraction Decomposition** The integral on the left side is complex due to the multiple factors in the denominator. To tackle this, we can use **partial fraction decomposition**, expressing the integrand as a sum of simpler fractions: $$ \frac{1}{y (y - 1)^2 (y - 2)^3 (y - 3)^4} = \frac{A}{y} + \frac{B}{y - 1} + \frac{C}{(y - 1)^2} + \frac{D}{y - 2} + \frac{E}{(y - 2)^2} + \frac{F}{(y - 2)^3} + \frac{G}{y - 3} + \frac{H}{(y - 3)^2} + \frac{I}{(y - 3)^3} + \frac{J}{(y - 3)^4} $$ Finding the coefficients $$ A, B, C, \ldots, J $$ involves solving a system of equations obtained by equating the coefficients of corresponding powers of $$ y $$ on both sides. This process can be tedious but is straightforward with systematic algebraic manipulation or using computational tools. ### 4. **Integrate Term by Term** Once the partial fractions are determined, integrate each term individually: $$ \int \left( \frac{A}{y} + \frac{B}{y - 1} + \frac{C}{(y - 1)^2} + \cdots + \frac{J}{(y - 3)^4} \right) dy = A \ln |y| + B \ln |y - 1| - \frac{C}{y - 1} + D \ln |y - 2| - \frac{E}{y - 2} + \frac{F}{2(y - 2)^2} + \cdots + \text{constant terms} + C $$ Each integral follows standard forms, such as: - $$ \int \frac{1}{y - a} \, dy = \ln |y - a| + C $$ - $$ \int \frac{1}{(y - a)^n} \, dy = \frac{1}{1 - n} \cdot \frac{1}{(y - a)^{n-1}}} + C $$ for $$ n \neq 1 $$ ### 5. **Express the General Solution** After integrating, the general solution will be an implicit equation involving logarithmic and rational functions of $$ y $$: $$ A \ln |y| + B \ln |y - 1| - \frac{C}{y - 1} + D \ln |y - 2| - \frac{E}{y - 2} + \frac{F}{2(y - 2)^2} + \cdots = t + C $$ Solving explicitly for $$ y $$ in terms of $$ t $$ may not be feasible due to the complexity of the equation. Therefore, the solution is often left in this implicit form or expressed using special functions if applicable. ### 6. **Qualitative Analysis (Optional)** Beyond finding an explicit solution, understanding the behavior of solutions can be insightful: - **Equilibrium Points:** $$ y = 0, 1, 2, 3 $$ are equilibrium solutions since $$ dy/dt = 0 $$ at these points. - **Stability:** - **$$ y = 0 $$:** Unstable (source). Solutions move away from 0 as $$ t $$ increases. - **$$ y = 1 $$:** Semi-stable. Solutions approach 1 from one side and move away on the other. - **$$ y = 2 $$:** Unstable (source). Solutions move away from 2 as $$ t $$ increases. - **$$ y = 3 $$:** Stable (sink). Solutions approach 3 as $$ t $$ increases. - **Behavior Between Equilibria:** - For $$ y < 0 $$: $$ dy/dt > 0 $$ (increasing) - For $$ 0 < y < 1 $$: $$ dy/dt < 0 $$ (decreasing) - For $$ 1 < y < 2 $$: $$ dy/dt < 0 $$ (decreasing) - For $$ 2 < y < 3 $$: $$ dy/dt > 0 $$ (increasing) - For $$ y > 3 $$: $$ dy/dt > 0 $$ (increasing) ### 7. **Conclusion** The differential equation is separable and can be solved implicitly using partial fraction decomposition. However, the explicit solution in terms of elementary functions is complex and typically expressed implicitly. For practical purposes, numerical methods or qualitative analysis are often employed to understand the behavior of solutions. The differential equation $$ \frac{dy}{dt} = y (y - 1)^2 (y - 2)^3 (y - 3)^4 $$ can be solved by separating variables and using partial fraction decomposition. After integrating, the general solution is expressed implicitly as: $$ A \ln |y| + B \ln |y - 1| - \frac{C}{y - 1} + D \ln |y - 2| - \frac{E}{y - 2} + \frac{F}{2(y - 2)^2} + \cdots = t + C $$ This equation cannot be solved explicitly for $$ y $$ in terms of $$ t $$ using elementary functions. Therefore, the solution is typically left in its implicit form or analyzed qualitatively to understand the behavior of solutions.

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Solution

1. Separate Variables

2. Integrate Both Sides

3. Partial Fraction Decomposition

4. Integrate Term by Term

5. Express the General Solution

6. Qualitative Analysis (Optional)

7. Conclusion

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