*dtdQ*​=35−4*Qβ* Now take β=1.7*β*=1.7 and determine the new equilibrium solution QS*QS*​ of Equation (2)(2).

QS=*QS*​=

To solve for the new equilibrium quantity \( Q_S \) in the given differential equation, let's proceed step by step.

The equation provided is:

\[
\frac{dQ}{dt} = 35 - 4 Q^\beta
\]

where \( \beta = 1.7 \).

The equilibrium solution occurs when the system reaches a steady state, meaning the change in \( Q \) with respect to time becomes zero:

\[
\frac{dQ}{dt} = 0
\]

So, we set the equation to 0 and solve for \( Q_S \):

\[
0 = 35 - 4 Q_S^{1.7}
\]

Rearranging the equation:

\[
4 Q_S^{1.7} = 35
\]

Now, solve for \( Q_S \):

\[
Q_S^{1.7} = \frac{35}{4} = 8.75
\]

Take the power of \( \frac{1}{1.7} \) on both sides to isolate \( Q_S \):

\[
Q_S = \left( 8.75 \right)^{\frac{1}{1.7}}
\]

Now, let's compute the numerical value for \( Q_S \).The equilibrium solution \( Q_S \) is approximately 3.58.

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. How does changing the value of \( \beta \) affect the equilibrium solution?
2. What happens to the system's dynamics if the constant 35 is altered?
3. How can we interpret this equilibrium in a real-world context (e.g., in economics or biology)?
4. Can the system have multiple equilibrium points under different conditions?
5. What if we introduce a time-dependent factor to the equation?

**Tip:** In nonlinear equations like this one, equilibrium points give insight into the long-term behavior of the system. Always check their stability to fully understand the system's dynamics.

The query submitted by the user: dtdQ = 35 − 4 Q^β. Now take β = 1.7 and determine the new equilibrium solution QS of Equation (2). QS =

Explore how to solve the nonlinear differential equation dQ/dt = 35 - 4 Q^β with β = 1.7, to find the equilibrium solution QS. This step-by-step solution focuses on key concepts in nonlinear systems and equilibrium behavior, suitable for undergraduate students.

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