# ordinary differential equations – finding a particular solution for inhomogeneous DE: \$ y & # 39; & # 39; – y + + frac {1} {4} y = 3 + e ^ frac {x} {2} \$

I have the inhomogeneous differential equation:

$$y & # 39; & # 39; – y + + frac {1} {4} y = 3 + e ^ frac {x} {2}$$

I start by finding the specific solution, and here I am stuck. At first I know that I have to $$y_p = y_ {p1} + y_ {p2}$$

From where $$y_ {p1}$$ will be in shape $$Ax ^ 2 + Bx + C$$ and $$y_ {p2}$$ will be in shape $$De ^ { frac {x} {2}}$$

$$y_ {p2} & # 39; = frac {1} {2} De ^ { frac {x} {2}}$$

$$y_ {p2} & # 39; & # 39; = frac {1} {4} De ^ { frac {x} {2}}$$

Connect this to the above differential equation and look for it $$e ^ { frac {x} {2}}$$ on the right side I get:

$$frac {1} {4} D – frac {1} {2} D + frac {1} {4} D = 1$$

$$0D = 1$$?

Where do I go from here?