These two problems have stumped me Im wondering which proced

These two problems have stumped me. I\'m wondering which procedure I am supposed to follow for these types of equations? Any help is much appreciated.

Solve the given differential equations by undetermined coeffients and the annihilator approach.

1) y\'\'+y\'=8

and

2) y\'\'+4y\'+4y=5x+7

Solution

I am solving the first problem, please post one more problem to get the second part answer. Thanks, Nikhil

1) y\'\' + y\' = 8

First we need to write the characteristic equation, for characteristic equation y\'\' coefficient will come with p^2, y\' with p and y term will come as it is

characteristics equation of above polynomial

p^2 + p = 0 or p(p+1) = 0

p=0 or p=-1

y(x) = c1*e^(-1x) + c2*e^(0x) = c1*e^(-x) + c2

Now we need to solve for the yp (particular solution), let us assume the particular solution is of the form of

yp = Ax^2 + Bx + C

yp\' = 2Ax + B

yp\'\' = 2A

substituting in the equation

2A + (2Ax+B) = 8

now comparing the coefficients A=0 & 2A+B = 8

which implies A=0,B=8 and C=0

Hence particular solution us yp = 8x

Final solution is y = yh + yp = c1*e^(-x) + c2 + 8x