Of undetermined coefficients to find a general solution of y

Of undetermined coefficients to find a general solution of: y\' + 2y\' + y = 3e^-x + 2x

Solution

Given ODE

y\'\' + 2y\' + y = 3e^-x + 2x ..............(*)

so the general solution will be, y = y_h + y_p

to find y_h,

we write the characteristic eqn,

m² + 2m + m = 0

(m+1)² = 0

m=-1,-1

hence,

y_h = c₁e^-x + c₂xe^-x

now to find y_p,

let y_p = Ax + B + Cx²e^-x { we cant take e^-x.Since, e^-x is in the homogeneous eqn already hence taking x2e^-x}

y_p\' = A + 2Cxe^-x - Cx²e^-x

y_p\'\'= 2Ce^-x -2Cxe^-x -2Cxe^-x + Cx²e^-x

putting these values in (*),we get

2Ce^-x - 2Cxe^-x - 2Cxe^-x + Cx²e^-x + 2A + 4Cxe^-x - 2Cx²e^-x + Ax + B + Cx²e^-x = 2x + 3e^-x

equating the coefficient of x & e^-x and the constant term,we get

A=2

B=-4

C=3/2

hence the general equation,

i.e..

y= y_h + y_p

= c₁e^-x + c₂xe^-x +3/2x²e^-x + 2x - 4