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Question-43822

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Question Number 43822 by peter frank last updated on 15/Sep/18
Answered by Joel578 last updated on 17/Sep/18
(b) 4cosh x + 3cosh y = 10 ...(i) 4sinh x + 3sinh y = 7 ...(ii) 16cosh^2 x + 24 cosh x cosh y + 9cosh^2 y = 100 ...(iii) 16sinh^2 x + 24 sinh x sinh y + 9sinh^2 y = 49 ...(iv) (iii) − (iv) 16(cosh^2 x − sinh^2 x) + 24(cosh x cosh y − sinh x sinh y) + 9(cosh^2 y − sinh^2 y) = 51 16 + 24 cosh (x − y) + 9 = 51 cosh (x − y) = ((26)/(24)) = ((13)/(12)) ((e^(x−y) + e^(y−x) )/2) = ((13)/(12)) (e^x /e^y ) + (e^y /e^x ) = ((13)/6) (i) + (ii) 4(cosh x + sinh x) + 3(cosh y + sinh y) = 17 4e^x + 3e^y = 17 Let e^x = a, e^y = b (a/b) + (b/a) = ((13)/6) → ((a^2 + b^2 )/(ab)) = ((13)/6) → a^2 + b^2 = ((13)/6)ab 4a + 3b = 17 → a = ((17 − 3b)/4) (((17 − 3b)/4))^2 + b^2 = ((13)/6)(((17 − 3b)/4))b ((289 − 102b + 25b^2 )/2) = ((221b − 39b^2 )/3) 3(289 − 102b + 25b^2 ) = 2(221b − 39b^2 ) 153b^2 − 748b + 867 = 0 9b^2 − 44b + 51 = 0 b = 3 ∨ b = ((17)/9) b = 3 → a = 2 ⇒ x = ln 2, y = ln 3 b = ((17)/9) → a = ((17)/6) ⇒ x = ln (((17)/6)), y = ln (((17)/9))
$$\left({b}\right) \\ $$$$\mathrm{4cosh}\:{x}\:+\:\mathrm{3cosh}\:{y}\:=\:\mathrm{10}\:\:\:\:…\left({i}\right) \\ $$$$\mathrm{4sinh}\:{x}\:+\:\mathrm{3sinh}\:{y}\:=\:\mathrm{7}\:\:\:\:\:\:\:\:…\left({ii}\right) \\ $$$$ \\ $$$$\mathrm{16cosh}^{\mathrm{2}} \:{x}\:+\:\mathrm{24}\:\mathrm{cosh}\:{x}\:\mathrm{cosh}\:{y}\:+\:\mathrm{9cosh}^{\mathrm{2}} \:{y}\:=\:\mathrm{100}\:\:\:…\left({iii}\right)\:\:\:\: \\ $$$$\mathrm{16sinh}^{\mathrm{2}} \:{x}\:+\:\mathrm{24}\:\mathrm{sinh}\:\:{x}\:\mathrm{sinh}\:\:{y}\:+\:\mathrm{9sinh}^{\mathrm{2}} \:{y}\:=\:\mathrm{49}\:\:\:\:…\left({iv}\right) \\ $$$$ \\ $$$$\left({iii}\right)\:−\:\left({iv}\right) \\ $$$$\mathrm{16}\left(\mathrm{cosh}^{\mathrm{2}} \:{x}\:−\:\mathrm{sinh}^{\mathrm{2}} \:{x}\right)\:+\:\mathrm{24}\left(\mathrm{cosh}\:{x}\:\mathrm{cosh}\:{y}\:−\:\mathrm{sinh}\:\:{x}\:\mathrm{sinh}\:\:{y}\right)\:+\:\mathrm{9}\left(\mathrm{cosh}^{\mathrm{2}} \:{y}\:−\:\mathrm{sinh}^{\mathrm{2}} \:{y}\right)\:=\:\mathrm{51} \\ $$$$\mathrm{16}\:+\:\mathrm{24}\:\mathrm{cosh}\:\left({x}\:−\:{y}\right)\:+\:\mathrm{9}\:=\:\mathrm{51} \\ $$$$\mathrm{cosh}\:\left({x}\:−\:{y}\right)\:=\:\frac{\mathrm{26}}{\mathrm{24}}\:=\:\frac{\mathrm{13}}{\mathrm{12}} \\ $$$$\frac{{e}^{{x}−{y}} \:+\:{e}^{{y}−{x}} }{\mathrm{2}}\:=\:\frac{\mathrm{13}}{\mathrm{12}} \\ $$$$\frac{{e}^{{x}} }{{e}^{{y}} }\:+\:\frac{{e}^{{y}} }{{e}^{{x}} }\:=\:\frac{\mathrm{13}}{\mathrm{6}} \\ $$$$ \\ $$$$\left({i}\right)\:+\:\left({ii}\right) \\ $$$$\mathrm{4}\left(\mathrm{cosh}\:{x}\:+\:\mathrm{sinh}\:{x}\right)\:+\:\mathrm{3}\left(\mathrm{cosh}\:{y}\:+\:\mathrm{sinh}\:{y}\right)\:=\:\mathrm{17} \\ $$$$\mathrm{4}{e}^{{x}} \:+\:\mathrm{3}{e}^{{y}} \:=\:\mathrm{17} \\ $$$$ \\ $$$$\mathrm{Let}\:{e}^{{x}} \:=\:{a},\:\:{e}^{{y}} \:=\:{b} \\ $$$$\frac{{a}}{{b}}\:+\:\frac{{b}}{{a}}\:=\:\frac{\mathrm{13}}{\mathrm{6}}\:\:\rightarrow\:\:\frac{{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} }{{ab}}\:=\:\frac{\mathrm{13}}{\mathrm{6}}\:\rightarrow\:{a}^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\frac{\mathrm{13}}{\mathrm{6}}{ab} \\ $$$$\:\mathrm{4}{a}\:+\:\mathrm{3}{b}\:=\:\mathrm{17}\:\:\rightarrow\:\:{a}\:=\:\frac{\mathrm{17}\:−\:\mathrm{3}{b}}{\mathrm{4}} \\ $$$$ \\ $$$$\left(\frac{\mathrm{17}\:−\:\mathrm{3}{b}}{\mathrm{4}}\right)^{\mathrm{2}} \:+\:{b}^{\mathrm{2}} \:=\:\frac{\mathrm{13}}{\mathrm{6}}\left(\frac{\mathrm{17}\:−\:\mathrm{3}{b}}{\mathrm{4}}\right){b} \\ $$$$\frac{\mathrm{289}\:−\:\mathrm{102}{b}\:+\:\mathrm{25}{b}^{\mathrm{2}} }{\mathrm{2}}\:=\:\frac{\mathrm{221}{b}\:−\:\mathrm{39}{b}^{\mathrm{2}} }{\mathrm{3}} \\ $$$$\mathrm{3}\left(\mathrm{289}\:−\:\mathrm{102}{b}\:+\:\mathrm{25}{b}^{\mathrm{2}} \right)\:=\:\mathrm{2}\left(\mathrm{221}{b}\:−\:\mathrm{39}{b}^{\mathrm{2}} \right) \\ $$$$\mathrm{153}{b}^{\mathrm{2}} \:−\:\mathrm{748}{b}\:+\:\mathrm{867}\:=\:\mathrm{0} \\ $$$$\mathrm{9}{b}^{\mathrm{2}} \:−\:\mathrm{44}{b}\:+\:\mathrm{51}\:=\:\mathrm{0} \\ $$$${b}\:=\:\mathrm{3}\:\:\vee\:\:{b}\:=\:\frac{\mathrm{17}}{\mathrm{9}} \\ $$$$ \\ $$$${b}\:=\:\mathrm{3}\:\:\rightarrow\:\:{a}\:=\:\mathrm{2}\: \\ $$$$\Rightarrow\:{x}\:=\:\mathrm{ln}\:\mathrm{2},\:\:{y}\:=\:\mathrm{ln}\:\mathrm{3} \\ $$$$ \\ $$$${b}\:=\:\frac{\mathrm{17}}{\mathrm{9}}\:\:\rightarrow\:\:{a}\:=\:\frac{\mathrm{17}}{\mathrm{6}} \\ $$$$\Rightarrow\:{x}\:=\:\mathrm{ln}\:\left(\frac{\mathrm{17}}{\mathrm{6}}\right),\:\:{y}\:=\:\mathrm{ln}\:\left(\frac{\mathrm{17}}{\mathrm{9}}\right) \\ $$
Commented by peter frank last updated on 16/Sep/18
find x and y
$${find}\:{x}\:{and}\:{y} \\ $$

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