Question Number 103138 by bemath last updated on 13/Jul/20

(D^2 −4D)y = x^2  e^(2x)

Commented bybemath last updated on 14/Jul/20

Commented byIsrarchajk724 last updated on 13/Jul/20

    please any one tell me can we convert math editor to word ...?

Answered by mathmax by abdo last updated on 14/Jul/20

y^(′′) −4y^′  =x^2  e^(2x)   (h)→r^2 −4r =0 ⇒r(r−4)=0 ⇒r =o or r=4 ⇒y_h =a +be^(4x)   =au_1  +bu_2   W(u_1 ,u_2 ) = determinant (((1           e^(4x) )),((0           4e^(4x) )))=4e^(4x)  ≠0  W_1 = determinant (((o          e^(4x) )),((x^2  e^(2x)    4e^(4x) )))=−x^2  e^(6x)   W_2 = determinant (((1                  o)),((1                x^2  e^(2x) )))=x^2  e^(2x)   v_1 =∫ (w_1 /W)dx =∫  ((−x^2  e^(6x) )/(4e^(4x) ))dx =−(1/4)∫x^2   e^(2x)  dx  ∫x^2  e^(2x)  dx =(x^2 /2)e^(2x)  −∫ 2x×(1/2)e^(2x)  dx =(x^2 /2)e^(2x) −∫ xe^(2x)  dx  =(x^2 /2)e^(2x) −{ (x/2) e^(2x)  −∫ (1/2)e^(2x) dx} =(x^2 /2)e^(2x) −(x/2)e^(2x)  +(1/4)e^(2x)  ⇒  v_1 =−(1/4){(x^2 /2)e^(2x)  −(x/2)e^(2x) +(1/4)e^(2x) }  v_2 =∫ (w_2 /w)dx =∫ ((x^2  e^(2x) )/(4e^(4x) ))dx =(1/4) ∫ x^2  e^(−2x)  dx  4v_2 =−(x^2 /2)e^(−2x) −∫ 2x×(−(1/2))e^(−2x) dx =−(x^2 /2)e^(−2x) +∫ xe^(−2x)  dx  =−(x^2 /2)e^(−2x)  −(x/2)e^(−2x)  +∫(1/2)e^(−2x) dx =−(x^2 /2)e^(−2x) −(x/2)e^(−2x) −(1/4)e^(−2x)  ⇒  v_2 =(1/4){ −(x^2 /2)−(x/2)−(1/4)}e^(−2x)   y_p =u_1 v_1  +u_2 v_2 =−(1/4){(x^2 /2)−(x/2) +(1/4)}e^(2x)  +e^(4x) ×(1/4){−(x^2 /2)−(x/2)−(1/4)}e^(−2x)   =(1/4){−(x^2 /2) +(x/2)−(1/4)}e^(2x)   +(1/4){−(x^2 /2)−(x/2)−(1/4)}e^(2x)   =(1/4){−x^2 −(1/2)}e^(2x)  =−(1/4)(x^2  +(1/2))e^(2x)   y =y_(h )  +y_p