y''+4y'+4y=(e^(-2x)) / (x^2), x>0 Determine a particular solution of the nonhomogeneous differential equation using the method of variation of parameters.

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beckden | High School Teacher | (Level 1) Educator

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`lambda^2+4lambda+4=0`

Solution is `(lambda+2)^2 =0`

`lambda = -2`

We then obtain `u_1=e^(-2x)` and `u_2 = xe^(-2x)`

The Wronskian is

`W=-e^(-2x)e^(-2x)(2x-1)+2xe^(-4x)=e^(-4x)`

Since the Wronksian is non zero the two functions are independent and therefore the two functions are the general solution to the differential equation.

We find the particular solution by computing `A(x)u_1+B(x)u_2`

`A(x)=-int 1/Wu_2(x)b(x) dx = -int 1/e^(-4x) xe^(-2x) e^(-2x)/x^2 dx=-int 1/x dx`

So `A(x) = -ln(x)+C_1`

`B(x) = int 1/W u_1(x)b(x) dx = int 1/e^(-4x)e^(-2x)e^(-2x)/x^2 dx = int 1/x^2 dx`

So `B(x)=-1/x + C_2`

So our general solution is

`y = (-ln(x)+C_1)e^(-2x)+(-1/x+C_2)xe^(-2x)` Simplifying

`y = -(ln(x)+1)e^(-2x) + C_1e^(-2x) + C_2xe^(-2x)`

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