Question

The resquest of the y'+6y=e^4t, y(0)=2

Answer 1

This is a linear differential equation of first order. Solve this by integrating the coefficient of the y term and then raising e to the integrated coefficient to find the integrating factor, i.e. the integrating factor for this problem is e^(6x). 
Multiplying both sides of the equation by the integrating factor: 

(y')e^(6x) + 6ye^(6x) = e^(12x) 

The left side is the derivative of ye^(6x), hence 

d/dx[ye^(6x)] = e^(12x) 

Integrating 

ye^(6x) = (1/12)e^(12x) + c where c is a constant 

y = (1/12)e^(6x) + ce^(-6x) 

Use the initial condition y(0)=-8 to find c: 

-8 = (1/12) + c 
c=-97/12 

Hence 

y = (1/12)e^(6x) - (97/12)e^(-6x)