Green's function and differential equation

Question

By using the Green's function, solve:

y''-αy =e^(-αx) with initial conditions y(0)=y'(0) = 0

thank you to whoever can do this in detail for me :)

Answer 1

We have here a 2nd order differential equation because we have a second derivative term.  In addition, it is non-homogenous.  Therefore, we need to find the general solution when it is homogenous (when the differential equation is equal to zero), and find the particular solution (equation not equal to zero).

 

y'' - ay = e^-ax    ;     y(0) = 0

                             y'(0) = 0

 

First step solve for homogenous y_H.

 

We let y= e^rx

          y'=re^rx

          y''=r²e^rx

 

Substituting those values into the equation, we obtain

 

r²e^rx - ae^rx = 0

 

By factoring out the e^rx term, we have the characteristic equation

 

r² - a = 0

r² = a

r = ±√(a)

 

The general solution will be in the form   y_H=C₁e^r₁x + C₂e^r₂x  where r₁ and r₂ are the roots of the characteristic equation.  In this case, our solution is

 

y_H = C₁e^x√(a) + C₂e^-x√(a)

 

 

Second step: solve for the particular solution y_P.  After that, combine y_H with y_P to get the final solution.

 

This time we let y_P = Be^-ax

                        y'_P = -aBe^-ax

                        y''_P = a²Be^-ax

 

Substitute these values into the non-homogenous equation, we have

 

a²Be^-ax - aBe^-ax =  e^-ax

 

e^-ax(a² - aB) = e^-ax

 

 

Now we equate coefficients to find B.  Keep in mind that a is a constant within the equation.

 

e^-ax :  a² - aB = 1

        

-aB = 1 - a²

 

B = -(1 - a²) / a

 

 

The particular solution is

 

y_P = [-(1 - a²) / a]e^-ax

 

 

Lastly, use the initial conditions to find C₁ and C₂ once you combine the solutions.