Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval $I$ of definition for the solution. $4y' + y = 0$ when $y = e^{-x/4}$

Neetesh Kumar | November 09, 2024

Differential Equation Homework Help

This is the solution to Math 2A, section 13Z, Fall 2023 | WebAssign
Math002ACh1Sec01 (Homework) Question - 4
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Step-by-Step-Solution:

Step 1: Compute $y'$

Given $y = e^{-x/4}$, find $y'$.

Since $y = e^{-x/4}$, differentiate with respect to $x$:

$y' = \boxed{-\frac{e^{-\frac{x}{4}}}{4} }$

Step 2: Substitute $y$ and $y'$ into the Differential Equation

Now substitute $y' = -\frac{1}{4} e^{-x/4}$ and $y = e^{-x/4}$ into the differential equation $4y' + y = 0$.

Calculation:

$4y' + y = 4\left(-\frac{1}{4} e^{-x/4}\right) + e^{-x/4}$

Simplify each term:

$4y' + y = \boxed{-e^{-\frac{x}{4}}} + e^{-x/4}$ $= \boxed{0}$

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