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Question Number 79423 by Rio Michael last updated on 25/Jan/20

solve for x and y sinh x − 2cosh y = 0 3cosh x + 6 sihn y = 5

$${solve}\:{for}\:{x}\:{and}\:{y}\: \\ $$$$\:\:\:{sinh}\:{x}\:−\:\mathrm{2}{cosh}\:{y}\:=\:\mathrm{0} \\ $$$$\:\:\:\mathrm{3}{cosh}\:{x}\:+\:\mathrm{6}\:{sihn}\:{y}\:=\:\mathrm{5} \\ $$

Answered by mr W last updated on 25/Jan/20

cosh y=((sinh x)/2) ...(i) sinh y=(5/6)−((cosh x)/2) ...(ii) (i)^2 −(ii)^2 : 1=((sinh^2 x)/4)−((25)/(36))+(5/6)cosh x−((cosh^2 x)/4) 1=−((25)/(36))+(5/6)cosh x−1 ⇒cosh x=((97)/(30)) ⇒x=cosh^(−1) ((97)/(30))=ln (((97)/(30))+(√((((97)/(30)))^2 −1 )))=ln ((97+(√(8509)))/(30)) sinh x=±(√((((97)/(30)))^2 −1))=±((√(8509))/(30)) cosh y=((sinh x)/2)=±((√(8509))/(60))>1 ⇒only + ⇒y=cosh^(−1) ((√(8509))/(60))=ln (((√(4909))+(√(8509)))/(60))

$$\mathrm{cosh}\:{y}=\frac{\mathrm{sinh}\:{x}}{\mathrm{2}}\:\:\:\:\:...\left({i}\right) \\ $$$$\mathrm{sinh}\:{y}=\frac{\mathrm{5}}{\mathrm{6}}−\frac{\mathrm{cosh}\:{x}}{\mathrm{2}}\:\:\:...\left({ii}\right) \\ $$$$\left({i}\right)^{\mathrm{2}} −\left({ii}\right)^{\mathrm{2}} : \\ $$$$\mathrm{1}=\frac{\mathrm{sinh}^{\mathrm{2}} \:{x}}{\mathrm{4}}−\frac{\mathrm{25}}{\mathrm{36}}+\frac{\mathrm{5}}{\mathrm{6}}\mathrm{cosh}\:{x}−\frac{\mathrm{cosh}^{\mathrm{2}} \:{x}}{\mathrm{4}} \\ $$$$\mathrm{1}=−\frac{\mathrm{25}}{\mathrm{36}}+\frac{\mathrm{5}}{\mathrm{6}}\mathrm{cosh}\:{x}−\mathrm{1} \\ $$$$\Rightarrow\mathrm{cosh}\:{x}=\frac{\mathrm{97}}{\mathrm{30}} \\ $$$$\Rightarrow{x}=\mathrm{cosh}^{−\mathrm{1}} \:\frac{\mathrm{97}}{\mathrm{30}}=\mathrm{ln}\:\left(\frac{\mathrm{97}}{\mathrm{30}}+\sqrt{\left(\frac{\mathrm{97}}{\mathrm{30}}\right)^{\mathrm{2}} −\mathrm{1}\:}\right)=\mathrm{ln}\:\frac{\mathrm{97}+\sqrt{\mathrm{8509}}}{\mathrm{30}} \\ $$$$\mathrm{sinh}\:{x}=\pm\sqrt{\left(\frac{\mathrm{97}}{\mathrm{30}}\right)^{\mathrm{2}} −\mathrm{1}}=\pm\frac{\sqrt{\mathrm{8509}}}{\mathrm{30}} \\ $$$$\mathrm{cosh}\:{y}=\frac{\mathrm{sinh}\:{x}}{\mathrm{2}}=\pm\frac{\sqrt{\mathrm{8509}}}{\mathrm{60}}>\mathrm{1}\:\Rightarrow{only}\:+ \\ $$$$\Rightarrow{y}=\mathrm{cosh}^{−\mathrm{1}} \:\frac{\sqrt{\mathrm{8509}}}{\mathrm{60}}=\mathrm{ln}\:\frac{\sqrt{\mathrm{4909}}+\sqrt{\mathrm{8509}}}{\mathrm{60}} \\ $$

Commented by Rio Michael last updated on 27/Jan/20

thanks sir,seems like hyperbolic functions don′t suit simultaneous equations

$${thanks}\:{sir},{seems}\:{like}\:{hyperbolic}\:{functions}\:{don}'{t} \\ $$$${suit}\:{simultaneous}\:{equations} \\ $$

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