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Chapter 6: Q. 32 (page 570)

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52.
32. $\frac{d y}{d x} = - 5 y, y (0) = 2$

Short Answer

Expert verified

The answer is $y (x) = 2 e^{- 5 x}$

Step by step solution

01

Step 1. Given information

We have been given $\frac{d y}{d x} = - 5 y, y (0) = 2$

02

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

The differential equation can be solved by antidifferentiating.

$鈭� \frac{1}{y} d y = 鈭� - 5 d x \ln |y| = - 5 x + C y = e^{- 5 x + C} y = A e^{- 5 x} (e^{C} = A)$

Now put $y (0) = 2$ ,

$2 = A e^{- 5 (0)} 2 = A$

So, the answer is $y (x) = 2 e^{- 5 x}$

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