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## Homework Statement

solve the IVP

y

^{''}-8y

^{'}+16y=0

y(0)=2

y

^{'}(0)=7

## Homework Equations

## The Attempt at a Solution

auxiliary equation:

r

^{2}-8r+16=0

(r-4)

^{2}=0

so we have r=4 with m=2

y(x)= c

_{1}e

^{4x}+c

_{2}xe

^{4x}

y(0)--> c

_{1}e

^{0}+ 0 = 2

**c**

_{1}=2now we have:

y(x)=2e

^{4x}+c

_{2}xe

^{4x}

take first derivative

y

^{'}(x) = 8e

^{4x}+c

_{2}e

^{4x}+4c

_{2}xe

^{4x}(product rule)

y

^{'}(0) ----> 8e

^{0}+c

_{2}e

^{0}+ 0 = 7

8+c

_{2}=7

**c**

_{2}=-1so our final equation is:

**y(x)=2e**

^{4x}-xe^{4x}Everything seems ok, but just wanted to run it by here to make sure it's all done correctly. thanks all!