Question Number 103014 by bobhans last updated on 12/Jul/20

y′′ −y = (e^x /(e^x  + e^(−x) )) .

Answered by bobhans last updated on 12/Jul/20

the characteristic equation of the  homogenous equation is γ^2 −1=0 ; γ=±1  y_h  = C_1 e^x  + C_2 e^(−x)    say y_1 (x)=e^x  →y_1 ′ = e^x   y_2 (x)= e^(−x)  →y_2 ′ = −e^(−x)   Wronskian W(y_1 ,y_2 ) =  determinant (((y_1    y_2 )),((y_1 ′  y_2 ′)))=   determinant (((e^x      e^(−x) )),((e^x   −e^(−x) )))=−2≠0  I_1 = ∫ ((e^x ((e^x /(e^x +e^(−x) ))))/(W(y_1 ,y_2 ))) dx = −(1/2)∫ (e^(2x) /(e^x +e^(−x) )) dx   = −(1/2)e^x  +(1/2)tan^(−1) (e^x )  I_2 = ∫ ((e^(−x) ((e^x /(e^x +e^(−x) ))))/(−2)) dx = −(1/2)∫ (dx/(e^x +e^(−x) ))  = −(1/2)tan^(−1) (e^x )   y_p  = −e^x I_2 + e^(−x) I_1  = (1/2)e^x  tan^(−1) (e^x )−  (1/2)+(1/2)e^(−x)  tan^(−1) (e^x )   complete solution   y = C_1 e^x +C_2 e^(−x) +(1/2)(e^x  + e^(−x) )tan^(−1) (e^x )−(1/2)

Commented bybramlex last updated on 12/Jul/20

coll joss

Answered by mathmax by abdo last updated on 12/Jul/20

y^(′′) −y =(e^x /(e^x  +e^(−x) ))  (he)→y^(′′) −y =0 →r^2 −1=0→r =+^− 1 ⇒y_h =a e^x  +be^(−x)  =au_1  +bu_2   W(u_1 ,u_2 ) = determinant (((e^x           e^(−x) )),((e^x          −e^(−x) ))) =−2 ≠0  W_1 = determinant (((o                              e^(−x) )),(((e^x /(e^x  +e^(−x) ))              −e^(−x) )))=−(1/(e^x  +e^(−x) ))  W_2 = determinant (((e^x              0)),((e^x             (e^x /(e^x  +e^(−x) ))))) =(e^(2x) /(e^x  +e^(−x) ))  v_1 =∫ (W_1 /W)dx =∫  (dx/(2(e^x  +e^(−x) )))  =_(e^x =t)    (1/2) ∫    (dt/(t(t +t^(−1) ))) =(1/2) ∫ (dt/(t^2  +1))  =(1/2)arctan(t) =(1/2)arctan(e^x )  v_2 =∫ (W_2 /W)dx  =−∫  (e^(2x) /(2( e^x  +e^(−x) )))dx =_(e^x =t)   −(1/2) ∫  (t^2 /(t(t+t^(−1) ))) dt  =−(1/2) ∫  (t/(t+t^(−1) ))dt =−(1/2)∫ (t^2 /(t^2  +1))dx =−(1/2)∫ ((t^2 +1−1)/(t^2  +1))dt  =−(1/2)t +(1/2)arctan(t) =−(e^x /2) +(1/2) arctan(e^x ) ⇒  y_p =u_1 v_1  +u_2 v_2 =e^x ((1/2) arctan(e^x ))+e^(−x) (−(e^x /2) +(1/2) arctan(e^x ))  =(e^x /2) arctan(e^x )−(1/2) +(e^(−x) /2)arctan(e^x )  =ch(x)arctan(e^x )−(1/2)  the general solution  is  y =y_h  +y_p =ae^x  +be^(−x)  +ch(x)arctan(e^x )−(1/2)