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Question Number 103479 by mohammad17 last updated on 15/Jul/20

by ussing laplace find cosh(2t)cos(4t) ?

$${by}\:{ussing}\:{laplace}\:{find}\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\:? \\ $$

Answered by Worm_Tail last updated on 15/Jul/20

L( cosh(2t)cos(4t))=L((((e^(2t) +e^(−2t) )/2))(((e^(i4t) +e^(−i4t) )/2))) L( cosh(2t)cos(4t))=L(((e^((2+i4)t) +e^((2−i4)t) +e^((−2+i4)t) +e^(−(2+i4)t) )/4)) L( cosh(2t)cos(4t))=L((e^((2+i4)t) /4))+L((e^((2−4i)t) /4))+L((e^((−2+4i)t) /4))+L((e^((−2−i4)t) /4)) L( cosh(2t)cos(4t))=(1/(4(s−2−4i)))+(1/(4(s−2+4i)))+(1/(4(s+2−4i)))+(1/(4(s+2+4i))) L( cosh(2t)cos(4t))=((s+2+4i+s−2−4i)/(4(s−(2+4i))(s+(2+4i))))+((s+2−4i+s−2+4i)/(4(s−(2−4i)(s+(2−4i)))) L( cosh(2t)cos(4t))=((2s)/(4(s−(2+4i))(s+(2+4i))))+((2s)/(4(s−(2−4i)(s+(2−4i)))) L( cosh(2t)cos(4t))=(s/(2(s^2 +12−16i)))+(s/(2(s^2 +12+16i))) L( cosh(2t)cos(4t))=((s^3 +12s+16is)/(2(s^2 +12−16i)))((+s^3 +12s−16is)/((s^2 +12+16i))) L( cosh(2t)cos(4t))=((2s^3 )/(2(s^4 +12s^2 +16i))((+24s)/(s^2 +12s^2 +144+12∙16i+12s^2 −16is^2 −12(16i)−256))) L( cosh(2t)cos(4t))=(s^3 /((s^4 +24s^2 −112))((+12s)/))

$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)={L}\left(\left(\frac{{e}^{\mathrm{2}{t}} +{e}^{−\mathrm{2}{t}} }{\mathrm{2}}\right)\left(\frac{{e}^{{i}\mathrm{4}{t}} +{e}^{−{i}\mathrm{4}{t}} }{\mathrm{2}}\right)\right) \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)={L}\left(\frac{{e}^{\left(\mathrm{2}+{i}\mathrm{4}\right){t}} +{e}^{\left(\mathrm{2}−{i}\mathrm{4}\right){t}} +{e}^{\left(−\mathrm{2}+{i}\mathrm{4}\right){t}} +{e}^{−\left(\mathrm{2}+{i}\mathrm{4}\right){t}} }{\mathrm{4}}\right) \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)={L}\left(\frac{{e}^{\left(\mathrm{2}+{i}\mathrm{4}\right){t}} }{\mathrm{4}}\right)+{L}\left(\frac{{e}^{\left(\mathrm{2}−\mathrm{4}{i}\right){t}} }{\mathrm{4}}\right)+{L}\left(\frac{{e}^{\left(−\mathrm{2}+\mathrm{4}{i}\right){t}} }{\mathrm{4}}\right)+{L}\left(\frac{{e}^{\left(−\mathrm{2}−{i}\mathrm{4}\right){t}} }{\mathrm{4}}\right) \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{\mathrm{1}}{\mathrm{4}\left({s}−\mathrm{2}−\mathrm{4}{i}\right)}+\frac{\mathrm{1}}{\mathrm{4}\left({s}−\mathrm{2}+\mathrm{4}{i}\right)}+\frac{\mathrm{1}}{\mathrm{4}\left({s}+\mathrm{2}−\mathrm{4}{i}\right)}+\frac{\mathrm{1}}{\mathrm{4}\left({s}+\mathrm{2}+\mathrm{4}{i}\right)} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{{s}+\mathrm{2}+\mathrm{4}{i}+{s}−\mathrm{2}−\mathrm{4}{i}}{\mathrm{4}\left({s}−\left(\mathrm{2}+\mathrm{4}{i}\right)\right)\left({s}+\left(\mathrm{2}+\mathrm{4}{i}\right)\right)}+\frac{{s}+\mathrm{2}−\mathrm{4}{i}+{s}−\mathrm{2}+\mathrm{4}{i}}{\mathrm{4}\left({s}−\left(\mathrm{2}−\mathrm{4}{i}\right)\left({s}+\left(\mathrm{2}−\mathrm{4}{i}\right)\right)\right.} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{\mathrm{2}{s}}{\mathrm{4}\left({s}−\left(\mathrm{2}+\mathrm{4}{i}\right)\right)\left({s}+\left(\mathrm{2}+\mathrm{4}{i}\right)\right)}+\frac{\mathrm{2}{s}}{\mathrm{4}\left({s}−\left(\mathrm{2}−\mathrm{4}{i}\right)\left({s}+\left(\mathrm{2}−\mathrm{4}{i}\right)\right)\right.} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{{s}}{\mathrm{2}\left({s}^{\mathrm{2}} +\mathrm{12}−\mathrm{16}{i}\right)}+\frac{{s}}{\mathrm{2}\left({s}^{\mathrm{2}} +\mathrm{12}+\mathrm{16}{i}\right)} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{{s}^{\mathrm{3}} +\mathrm{12}{s}+\mathrm{16}{is}}{\mathrm{2}\left({s}^{\mathrm{2}} +\mathrm{12}−\mathrm{16}{i}\right)}\frac{+{s}^{\mathrm{3}} +\mathrm{12}{s}−\mathrm{16}{is}}{\left({s}^{\mathrm{2}} +\mathrm{12}+\mathrm{16}{i}\right)} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{\mathrm{2}{s}^{\mathrm{3}} }{\mathrm{2}\left({s}^{\mathrm{4}} +\mathrm{12}{s}^{\mathrm{2}} +\mathrm{16}{i}\right.}\frac{+\mathrm{24}{s}}{\left.{s}^{\mathrm{2}} +\mathrm{12}{s}^{\mathrm{2}} +\mathrm{144}+\mathrm{12}\centerdot\mathrm{16}{i}+\mathrm{12}{s}^{\mathrm{2}} −\mathrm{16}{is}^{\mathrm{2}} −\mathrm{12}\left(\mathrm{16}{i}\right)−\mathrm{256}\right)} \\ $$$${L}\left(\:{cosh}\left(\mathrm{2}{t}\right){cos}\left(\mathrm{4}{t}\right)\right)=\frac{{s}^{\mathrm{3}} }{\left({s}^{\mathrm{4}} +\mathrm{24}{s}^{\mathrm{2}} −\mathrm{112}\right.}\frac{+\mathrm{12}{s}}{\left.\right)} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\: \\ $$$$ \\ $$

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