Answer:
Option A.
Explanation:
This is a second order DE, so we'll need to integrate twice, applying initial conditions as we go. At a couple points, we'll need to apply u-substitution.
<u>Round 1:</u>
To solve the differential equation, write it as differentials, move the differential, and integrate both sides:
Applying various properties of integration:
Prepare for integration by u-substitution
, letting and
Find dt in terms of
Using the Exponential rule (don't forget your constant of integration):
Back substituting for :
<u>Finding the constant of integration</u>
Given initial condition
The first derivative with the initial condition applied:
<u>Round 2:</u>
Integrate again:
<u />
<u>Finding the constant of integration :</u>
Given initial condition
So,
<u>Checking the solution</u>
This matches our initial conditions here
Going back to the function, differentiate:
Apply Exponential rule and chain rule, then power rule
This matches our first order step and the initial conditions there.
Going back to the function y', differentiate:
Applying the Exponential rule and chain rule, then power rule
So our proposed solution is a solution to the differential equation, and satisfies the initial conditions given.