nonhomogenous second order differential equation ?

Kimmoh

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Nov 24, 2020
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Can someone help me to find the particular answer for the following differential equation, through undetermined coefficient ,
y `` + 3y` + 3y = (x + 1) e^x

Thanks in advanced.
 
Okay, the "non-homogeneous"part is \(\displaystyle (x+ 1)e^x\). That is precisely the kind of function we would expect as a solution to a homogeneous differential equation with 1 as a double root to the characteristic equation. So we can expect to get a solution to the entire equation by "undetermined coefficients", using a function of the form \(\displaystyle y= (Ax+ B)e^x\).

So what are y' and y'' for that y? Put those into the differential equation. You will have an equation of the form (Px+ Q)e^x on the right side (where P and Q are functions of A and B) and (x+ 1)e^x
Set P= 1 and Q= 1 and solve those two equations for A and B.
 
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