# Solve by Method of Variation of Parameters : ( D 2 − 6 D + 9 ) Y = E 3 X X 2 - Applied Mathematics 2

Sum

Solve by method of variation of parameters :(D^2-6D+9)y=e^(3x)/x^2

#### Solution

(D^2-6D+9)y=e^(3x)/x^2
For complementary solution ,
𝒇(𝑫)=𝟎

therefore(D^2-6D+9)=0

Roots are : D = 3 , 3 Real roots but repeatative.
The complementary solution of given diff. eqn is ,

therefore y_c=(c_1+xc_2)e^(3x)

For particular solution ,
By method of variation of parameters,

y_p=y_1p_1+y_2p_2

where p_1=int(-y_2X)/wdx

p_2=int(-y_1X)/wdx

w=|(y_1,y_2),(y'_1,y'_2)|

w=|(e^(3x),xe^(3x)),(3e^(3x),e^(3x)+3xe^(3x))|=e^(6x)

p_1=int(-y_2X)/wdx=int(xe^(3x))/e^(6x).e^(3x)/x^2dx=int(-1)/xdx=-logx

p_2=int(-y_1X)/wdx=int(e^(3x))/e^(6x).e^(3x)/x^2dx=int(1)/x^2dx=(-1)/x

The particular integral of given diff. eqn is given by,

thereforey_p=-e^(3x)logx-e^(3x)=-e^(3x)(logx+1)

The general solution of given diff. eqn is given by ,

y_g=y_c+y_p=(c_1+xc_2)e^(3x)=-e^(3x)(logx+1)

Concept: Method of Variation of Parameters
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