Bruchgleichungen mit Parametern
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Exercise:
Bestimme jeweils die Lösungsmenge L_x dieser Gleichungen mit Parametern ohne Diskussion der Sonderfälle. nprvmulticols abclist abc x + dfracxp abc dfracxe + dfracxf dfracf abc dfracn+x + dfracnx+ abc x - dfracdx c abc dfracx-t - dfract abc dfrac aw - dfrac wa dfraca^x+aw abc dfracx^+a^x^ + dfracx+ax abc dfracx + dfracx+a+dfracx+a abc dfracax+bax-b - dfracax-bax+b dfracaba^-b^ abc dfracx+a dfraca-bx dfraca+b dfracax+ abclist nprvmulticols
Solution:
newcommanddberensuremathD_x R setminus qty# newcommandxloesx # && L_x left # right newcommandloes&&L_# qty# abclist abc D_x R al x + dfracxp && | p xp+ p && |:p+ xloesfracpp+ abc D_x R al dfracxe + dfracxf dfracf &&| ef xf+e e &&|:f+e xloes fracee+f abc dber- al dfracn+x + dfracnx+ uf xx+ n+x+ + nx tu nx + n + x + + nx uf -n- nx + x tu xn+ -n- uf :n+ xloesfrac-n-n+ abc D_x R al x - dfracdx c &&| x-d c &&|:-d xloes fracc-d abc dbert al dfracx-t - dfract &&| tx-t t tx-t - x-t &&|text TU t tx - t^ -x +t &&| +t^ -t t^ xt- &&|:t- xloes fract^t- abc dber-aw al dfrac aw - dfrac wa dfraca^x+aw uf awx+aw a^x+aw - w^x+aw a^w tu a^x + a^w -w^x - aw^ a^w uf -a^w + aw^ a^-w^x aw^ uf :a^-w^ xloesfracaw^a^-w^ abc dber aldfracx^+a^x^ + dfracx+ax uf x^ x^ + a^ + xx+a x^ tu x^ + a^ + x^ + ax x^ tu x^ + ax + a^ x^ uf -x^ -x^ + ax + a^ mf x_ frac-a pm sqrta^ + a^- frac-a pm a- loesa -frac a abc dber-a-a aldfracx + dfracx+a+dfracx+a uf xx+ax+a x+ax+a + xx+a + xx+a tu x^ + ax + a^ + x^ + ax + x^ +ax tu x^ + ax + a^ mf x_ frac-a pm sqrta^ - a^ frac-a pm sqrta^ frac-a pm sqrta frac-pmsqrta loesfrac-+sqrta frac--sqrta abc dber-dfracba dfracba aldfracax+bax-b - dfracax-bax+b dfracaba^-b^ uf ax-bax+b ax+b^ -ax-b^ fracabax+bax-ba^-b^ tu abx fracaba^x^ - b^a^-b^ uf :ab a^-b^ a^-b^x a^x^-b^ uf -a^-b^x a^x^ -a^-b^x -b^ mf x_ fraca^-b^ pm sqrta^-b^^ + a^b^a^ fraca^-b^ pm a^+b^a^ loes-fracb^a^ abc dber- aldfracx+a dfraca-bx dfraca+b uf axa+bx+ x+^a-ba+b a a x tu x^ + x + a^-b^ a^ x tu a^-b^x^ + a^-b^x + a^ - b^ a^ uf -a^ a^-b^x^ - a^+b^x + a^ - b^ mf fraca^+b^ pm sqrta^+b^^ - a^-b^^a^-b^ x_ fraca^ + b^ pm aba+ba-b x_ fracapm ba+ba-b x_ loesfraca-b fraca+b abclist
Bestimme jeweils die Lösungsmenge L_x dieser Gleichungen mit Parametern ohne Diskussion der Sonderfälle. nprvmulticols abclist abc x + dfracxp abc dfracxe + dfracxf dfracf abc dfracn+x + dfracnx+ abc x - dfracdx c abc dfracx-t - dfract abc dfrac aw - dfrac wa dfraca^x+aw abc dfracx^+a^x^ + dfracx+ax abc dfracx + dfracx+a+dfracx+a abc dfracax+bax-b - dfracax-bax+b dfracaba^-b^ abc dfracx+a dfraca-bx dfraca+b dfracax+ abclist nprvmulticols
Solution:
newcommanddberensuremathD_x R setminus qty# newcommandxloesx # && L_x left # right newcommandloes&&L_# qty# abclist abc D_x R al x + dfracxp && | p xp+ p && |:p+ xloesfracpp+ abc D_x R al dfracxe + dfracxf dfracf &&| ef xf+e e &&|:f+e xloes fracee+f abc dber- al dfracn+x + dfracnx+ uf xx+ n+x+ + nx tu nx + n + x + + nx uf -n- nx + x tu xn+ -n- uf :n+ xloesfrac-n-n+ abc D_x R al x - dfracdx c &&| x-d c &&|:-d xloes fracc-d abc dbert al dfracx-t - dfract &&| tx-t t tx-t - x-t &&|text TU t tx - t^ -x +t &&| +t^ -t t^ xt- &&|:t- xloes fract^t- abc dber-aw al dfrac aw - dfrac wa dfraca^x+aw uf awx+aw a^x+aw - w^x+aw a^w tu a^x + a^w -w^x - aw^ a^w uf -a^w + aw^ a^-w^x aw^ uf :a^-w^ xloesfracaw^a^-w^ abc dber aldfracx^+a^x^ + dfracx+ax uf x^ x^ + a^ + xx+a x^ tu x^ + a^ + x^ + ax x^ tu x^ + ax + a^ x^ uf -x^ -x^ + ax + a^ mf x_ frac-a pm sqrta^ + a^- frac-a pm a- loesa -frac a abc dber-a-a aldfracx + dfracx+a+dfracx+a uf xx+ax+a x+ax+a + xx+a + xx+a tu x^ + ax + a^ + x^ + ax + x^ +ax tu x^ + ax + a^ mf x_ frac-a pm sqrta^ - a^ frac-a pm sqrta^ frac-a pm sqrta frac-pmsqrta loesfrac-+sqrta frac--sqrta abc dber-dfracba dfracba aldfracax+bax-b - dfracax-bax+b dfracaba^-b^ uf ax-bax+b ax+b^ -ax-b^ fracabax+bax-ba^-b^ tu abx fracaba^x^ - b^a^-b^ uf :ab a^-b^ a^-b^x a^x^-b^ uf -a^-b^x a^x^ -a^-b^x -b^ mf x_ fraca^-b^ pm sqrta^-b^^ + a^b^a^ fraca^-b^ pm a^+b^a^ loes-fracb^a^ abc dber- aldfracx+a dfraca-bx dfraca+b uf axa+bx+ x+^a-ba+b a a x tu x^ + x + a^-b^ a^ x tu a^-b^x^ + a^-b^x + a^ - b^ a^ uf -a^ a^-b^x^ - a^+b^x + a^ - b^ mf fraca^+b^ pm sqrta^+b^^ - a^-b^^a^-b^ x_ fraca^ + b^ pm aba+ba-b x_ fracapm ba+ba-b x_ loesfraca-b fraca+b abclist