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.