Q. 33 - Calculus

Question

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29–52.

33. $\frac{dy}{dx}=1-y,y\left(0\right)=4$

Step-by-Step Solution

Answer

The answer is $y\left(x\right)=3e^{-x}+1$

Step 1. Given information

We have been given $\frac{dy}{dx}=1-y,y\left(0\right)=4$

Step 2. Solve using antidifferentiation and/or variable separable method.

The differential equation can be solved by antidifferentiating.

$$\begin{gathered} \int \frac{1}{1-y}dy=\int dx \\ -\ln \left|1-y\right|=x+C \\ \ln \left|1-y\right|=-x-C \\ 1-y=e^{-x-C} \\ y=1-e^{-x-C} \\ y=1-A{e}^{-x} \left(e^{-C}=A\right) \end{gathered}$$

Now put $y\left(0\right)=4$,

$$\begin{gathered} 4=1-A{e}^{-\left(0\right)} \\ A=1-4 \\ A=-3 \end{gathered}$$

So, the answer is $y\left(x\right)=3e^{-x}+1$