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(Solved): Cooper Section 13.7 #1 a. What is the general solution of the equation y' = 3(1 - y)? (Use A as the ...



Cooper Section 13.7 #1 a. What is the general solution of the equation y' = 3(1 - y)? (Use A as the arbitrary constant.) b. What is the long term behaviour of solutions to this equation? (Draw diagrams to assist.) Oy1 as t→∞ y0 as t→ ∞ Oy-1 as t → ∞ Online Math Lab resources for this problem: • Differential Equations • Slope Fields

a. What is the general solution of the equation \( y^{\prime}=3(1-y) \) ? (Use \( A \) as the arbitrary constant.)
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y(t)=
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a. What is the general solution of the equation ? (Use as the arbitrary constant.) b. What is the long term behaviour of solutions to this equation? (Draw diagrams to assist.) Online Math Lab resources for this problem: - Differential Equations - Slope Fields


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a. To find the general solution of the given differential equation, we can separate the variables and integrate:
  
Separating variables:
  
Integrating both sides:
  
Using the substitution method, let's substitute   
  
Simplifying the integrals:
  
where C is the constant of integration. Substituting back u=1?y:
  
Taking the exponential of both sides:
  
Since the absolute value of a quantity can be positive or negative, we can rewrite this as two cases:

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