Solve the following differential equation with initial conditions: y''=e^-2t+10e^4t ; y(0)=1, y'(0)=0
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.
Round 1:
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 :
Finding the constant of integration
Given initial condition
The first derivative with the initial condition applied:
Round 2:
Integrate again:
Finding the constant of integration :
Given initial condition
So,
Checking the solution
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.
answer: the net deposit amount (option c)
in order to fill in the deposit slip, michael needs to fill in the amount that he wants to deposit in the bank. without the net deposit amount, he will not be able to deposit the money because it is mandatory for him to mention it, as the bank needs to maintain a record to avoid any inconveniences for the bank and michael in the future.
the bank officer checks and verifies the amount received against the amount written on the deposit slip to process the slip.