Homogeneousequation:(D2−2D+1)y=0λ2−2λ+1=0⇒λ1,2=1⇒⇒y=C1ex+C2xexu1(x)=ex,u2(x)=xexy(x)=A(x)u1(x)+B(x)u2(x)[u1(x)u1′(x)u2(x)u2′(x)][A′(x)B′(x)]=[0f(x)][exexxexex+xex][A′(x)B′(x)]=[0x3/2ex]⇒⇒[A′(x)B′(x)]=[exexxexex+xex]−1[0x3/2ex]==e−2x[ex(1+x)−ex−xexex][0x3/2ex]=[−x5/2x3/2]A(x)=−72x7/2,B=52x5/2y(x)=−72x7/2ex+52x7/2ex=−354x7/2ex−particularsolutiony(x)=−354x7/2ex+C1ex+C2xex−generalsolution
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