Chapter 6: Q. 11 (page 570)

In the process of solving the differential equation $\frac{d y}{d x} = 1 - y$ by separation of variables, we obtain the equation $- \ln | 1 - y | = x + C$ . After solving for $y$ , this equation becomes $y = 1 - A e^{- x}$ . Given that $y > 1$ , how is A related to $C$ ?

Short Answer

Expert verified

Ans: $A$ and $C$ are related by the relation $A = - e^{- C}$

Step by step solution

Step 1. Given information.

given equations,

$\frac{d y}{d x} = 1 - y$ , $- \ln | 1 - y | = x + C$ , $y = 1 - A e^{- x}$

Step 2. The differential equation dydx=1-y is solved by applying the variable separable method. After separating the variables and integrating both sides yields

$\begin{aligned} - \ln | 1 - y | = x + C \\ \ln | 1 - y | = - x - C \\ | 1 - y | = e^{- x - C} \end{aligned}$

Note that $y > 1$ , and the exponential function is always positive, So the above result leads to the conclusion

$\begin{aligned} y - 1 = e^{- x - c} \\ y = 1 + e^{- x - c} \\ = 1 + e^{- x} \cdot e^{- c} \end{aligned}$

Step 3.

In the above result, since $e^{- C}$ is a constant, replace it with another constant $A$ , so that the solution is written in compact form as role="math" localid="1649145545523" $y = 1 - A e^{- x}$ . Thus, the two constants $A$ and $C$ are related by the relation $A = - e^{- C}$

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In the process of solving the differential equation $\frac{d y}{d x} = 1 - y$ by separation of variables, we obtain the equation $- \ln | 1 - y | = x + C$ . After solving for $y$ , this equation becomes $y = 1 - A e^{- x}$ . Given that $y > 1$ , how is A related to $C$ ?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The differential equation dydx=1-y is solved by applying the variable separable method. After separating the variables and integrating both sides yields

Step 3.

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In the process of solving the differential equation dydx=1-yby separation of variables, we obtain the equation −ln⁡|1−y|=x+C. After solving for y, this equation becomes y=1−Ae−x. Given that y>1, how is A related to C?

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The differential equation dydx=1-y is solved by applying the variable separable method. After separating the variables and integrating both sides yields

Step 3.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Calculus

Mechanics Maths

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In the process of solving the differential equation $\frac{d y}{d x} = 1 - y$ by separation of variables, we obtain the equation $- \ln | 1 - y | = x + C$ . After solving for $y$ , this equation becomes $y = 1 - A e^{- x}$ . Given that $y > 1$ , how is A related to $C$ ?