Sure! Let's break down the steps for each equation: ## **Step-by-Step Guide** ### 1. **Find Equilibrium Solutions** Equilibrium solutions (also called critical points) are values where the derivative is zero. Set the right side of the equation to zero and solve for the variable. ### 2. **Phase Line Analysis** - Draw a number line. - Mark equilibrium points. - For each interval, check the sign of the derivative to determine the direction (increasing/decreasing). - Classify each equilibrium point: - **Sink**: Arrows point towards the equilibrium (stable). - **Source**: Arrows point away (unstable). - **Node**: Special cases, but for 1D, usually just sinks or sources. --- ## **Examples** ### **8. \( y' = 2y - 5 \)** #### **Equilibrium:** Set \( 2y - 5 = \implies y = 2.5 \) #### **Phase Line:** - For \( y < 2.5 \), \( y' < \) (decreasing) - For \( y > 2.5 \), \( y' > \) (increasing) #### **Classification:** - **Source** (arrows point away from \( y = 2.5 \)) --- ### **9. \( \frac{dx}{dt} = (x-1)(x+2) \)** #### **Equilibrium:** Set \( (x-1)(x+2) = \implies x = 1 \) or \( x = -2 \) #### **Phase Line:** - For \( x < -2 \): Both terms negative, product positive (\( x' > \)), arrow → - For \( -2 < x < 1 \): One term negative, product negative (\( x' < \)), arrow ← - For \( x > 1 \): Both terms positive, product positive (\( x' > \)), arrow → #### **Classification:** - At \( x = -2 \): **Source** (arrows away) - At \( x = 1 \): **Sink** (arrows toward) --- ### **10. \( \frac{dx}{dt} = (x^2-1)(x-2) \)** Factor: \( (x-1)(x+1)(x-2) \) #### **Equilibrium:** \( x = -1, 1, 2 \) #### **Phase Line:** - Check sign in each interval: - \( x < -1 \): \( -, -, - \) ⇒ \( - \) (\( x' < \)), ← - \( -1 < x < 1 \): \( +, -, - \) ⇒ \( + \) (\( x' > \)), → - \( 1 < x < 2 \): \( +, +, - \) ⇒ \( - \) (\( x' < \)), ← - \( x > 2 \): \( +, +, + \) ⇒ \( + \) (\( x' > \)), → #### **Classification:** - \( x = -1 \): **Source** (arrows away) - \( x = 1 \): **Sink** (arrows toward) - \( x = 2 \): **Source** (arrows away) --- ### **11. \( \frac{dy}{dx} = \sin(2y) \)** #### **Equilibrium:** \( \sin(2y) = \implies 2y = n\pi \implies y = n\pi/2 \), for any integer \( n \). #### **Phase Line:** - Between each equilibrium, use sign of \( \sin(2y) \) (periodic: +, -, +, ...) - For example, between \( y= \) and \( y=\pi/2 \), \( \sin(2y) > \) (increasing), next interval \( \sin(2y) < \) (decreasing). #### **Classification:** - Alternate between **sources** and **sinks** at each \( y = n\pi/2 \). --- ### **12. \( x' = (x^2+1)(x-1) \)** #### **Equilibrium:** - \( x^2 + 1 \) is always positive, so only \( x-1 = \implies x = 1 \) #### **Phase Line:** - For \( x < 1 \), \( x' < \) (decreasing) - For \( x > 1 \), \( x' > \) (increasing) #### **Classification:** - \( x = 1 \): **Source** (arrows away) --- ### **13. \( x' = x^2 + x + 1 \)** #### **Equilibrium:** - \( x^2 + x + 1 = \) has no real roots (discriminant < ) - **No equilibrium solutions** #### **Phase Line:** - \( x^2 + x + 1 > \) always, so \( x' > \) everywhere (always increasing) --- ## **Summary Table** | Equation | Equilibrium(s) | Classification | |----------|----------------|---------------| | 8 | \( y = 2.5 \) | Source | | 9 | \( x = -2, 1 \)| Source at -2, Sink at 1 | | 10 | \( x = -1, 1, 2 \) | Source at -1 & 2, Sink at 1 | | 11 | \( y = n\pi/2 \) | Alternate Source/Sink | | 12 | \( x = 1 \) | Source | | 13 | None | No equilibrium | If you want a diagram for any specific phase line, let me know!